Positive Integers n Such That σ(φ(n)) = σ(n)

Size: px
Start display at page:

Download "Positive Integers n Such That σ(φ(n)) = σ(n)"

Transcription

1 Journal of Integer Sequences, Vol. (2008), Article Positive Integers n Such That σ(φ(n)) = σ(n) Jean-Marie De Koninck Départment de Mathématiques Université Laval Québec, PQ GK 7P4 Canada Florian Luca Instituto de Matemáticas Universidad Nacional Autónoma de Méico C.P Morelia, Michoacán Méico Abstract In this paper, we investigate those positive integers n for which the equality σ(φ(n)) = σ(n) holds, where σ is the sum of the divisors function and φ is the Euler function. Introduction For a positive integer n we write σ(n) and φ(n) for the sum of divisors function and for the Euler function of n, respectively. In this note, we study those positive integers n such that σ(φ(n)) = σ(n) holds. This is sequence A03363 in Sloane s Online Encylopedia of Integer Sequences. Let A be the set of all such positive integers n and for a positive real number we put A() = A [,]. Our result is the following.

2 Theorem. The estimate holds for all real numbers >. ( ) #A() = O (log ) 2 The above upper bound might actually be the correct order of magnitude of #A(). Indeed, note that if m is such that σ(2φ(m)) = 2σ(m) () and if q > m is a Sophie Germain prime, that is a prime number q such that p = 2q + is also prime, then n = mp A. The numbers m = 238, 2806, 5734, 5937, 798, 8097,... all satisfy relation (), and form Sloane s sequence A More generally, if k and m are positive integers such that σ(2 k φ(m)) = 2 k σ(m) (2) and q <... < q k are primes with p i = 2q i + also primes for i =,...,k and q > m, then n = p...p k m A. Now recall that the Prime K-tuplets Conjecture of Dickson (see, for instance, [2, 4, 8]) asserts that, ecept in cases ruled out by obvious congruence conditions, K linear forms a i n+b i, i =,...,K, take prime values simultaneously for about c/(log ) K integers n, where c is a positive constant which depends only on the given linear forms. Under this conjecture (applied with K = 2 and the linear forms n and 2n+), we obtain that there should be /(log ) 2 Sophie Germain primes q, which suggests that #A() /(log ) 2. We will come back to the Sophie Germain primes later. Throughout, we use the Vinogradov symbols and and the Landau symbols O and o with their regular meanings. We use log for the natural logarithm and p, q and r with or without subscripts for prime numbers. 2 Preliminary Results In this section, we point out a subset B() of all the positive integers n of cardinality O(/(log ) 2 ). For the proof of Theorem we will work only with the positive integers n A()\B(). Further, 0 is a sufficiently large positive real number, where the meaning of sufficiently large may change from a line to the net. We put ( ) log y = ep. log log For a positive integer n we write P(n) for the largest prime factor of n. It is well known that Ψ(,y) = #{n P(n) y} = ep( ( + o())u log u) (u ), (3) where u = log / log y, provided that u y /2 (see [], Corollary.3 of [6], or Chapter III.5 of [9]). In our case, u = log log, so, in particular, the condition u y /2 is satisfied for > 0. We deduce that u log u = (log log )(log log log ). 2

3 Thus, if we set B () = {n P(n) y}, then We now let #B () = Ψ(,y) = ep ( ( + o())(log log )(log log log )) < for > (log ) 0 0. (4) z = (log ) 26 and for a positive integer n we write ρ(n) for its largest squarefull divisor. Recall that a positive integer m is squarefull if p 2 m whenever p is a prime factor of m. It is well known that if we write S(t) = {m t m is squarefull}, then #S(t) = ζ(3/2) ζ(3) t/2 + O(t /3 ), where ζ is the Riemann Zeta-Function (see, for eample, Theorem 4.4 in [7]). By partial summation, we easily get that m t/2. (5) m t m squarefull We now let B 2 () be the set of positive integers n such that one of the following conditions holds: (i) ρ(n) z, (ii) p n for some prime p such that ρ(p ± ) z, (iii) there eist primes r and p such that p n, p ± (mod r) and ρ(r ± ) z. We will find an upper bound for #B 2 (). Let B 2, () be the set of those n B 2 () for which (i) holds. We note that for every n B 2, () there eists a squarefull positive integer d z such that d n. For a fied d, the number of such n does not eceed /d. Hence, #B 2, () d z d squarefull d (log ) 3, (6) where we have used estimate (5) with t = z. Now let B 2,2 () be the set of those n B 2 () for which (ii) holds. We note that each n B 2,2 () has a prime divisor p such that p ± (mod d), where d is as above. Given d and p, the number of such n does not eceed /p. Summing up over all choices of p and d we get that #B 2,2 () d z d squarefull p ± (mod d) p (log log ) 2 d z d squarefull 3 p d d z d squarefull log log φ(d) (log log )2 (log ) 3, (7)

4 where in the above estimates we used aside from estimate (5), the fact that the estimate ( ) p log log p (a,b) + O (8) φ(b) p a (mod b) p holds uniformly in a, b and when b, where a and b are coprime and p (a,b) is the smallest prime number p a (mod b) (note that p (,b) b+ and p (,b) = p (b,b) b φ(b) for all b 2), together with the well known minimal order φ(n)/n / log log, valid for n in the interval [,]. Let B 2,3 () be the set of those n B 2 () for which (iii) holds. Then there eists r such that r ± (mod d) for some d as above, as well as p n such that r p or r p +. Given d, r and p, the number of such n does not eceed /p, and now summing up over all choices of d, r and p, we get that #B 2,3 () d z r ± (mod d) d squarefull r d z r ± (mod d) d squarefull r (log log ) 2 d z d squarefull (log log ) 3 d z d squarefull (log log ) 4 d z d squarefull p ± (mod r) p log log φ(r) r ± (mod d) r φ(d) d r p (log log )4 (log ) 3, (9) where in the above estimates we used again estimate (5), estimate (8) twice as well as the minimal order of the Euler function on the interval [,]. Hence, using estimates (6) (9), we get #B 2 () #B 2, () + #B 2,2 () + #B 2,3 () < (log ) 0 for > 0. (0) We now put and set w = 0 log log S(w,) = ω(m) w m m, 4

5 where ω(m) denotes the number of distinct prime factors of the positive integer m. Note that, using the fact that p = log log t + O() and Stirling s formula k! = (+o())kk e k 2πk, p t we have S(w,) = k w ω(m)=k m m = k w k! ( p α ) k p α = ( ( )) k k! p + O p 2 k w p p 2 k w ( ) k e log log + O() ( e log log + O() k w k w k w ( ) w e log log + O() w for > 0 because 0 log(0/e) >. (log log + O()))k k! ) k (log ) 0 log(0/e) < (log ) () We now let B 3 () be the set of positive integers n such that one of the following conditions holds: (i) ω(n) w, (ii) p n for some prime p for which ω(p ± ) w, (iii) there eist primes r and p such that p n, p ± (mod r) and ω(r ± ) w. Let B 3, (), B 3,2 () and B 3,3 () be the sets of n B 3 () for which (i), (ii) and (iii) hold, respectively. To bound the cardinality of B 3, (), note that, using (), we have #B 3, () = ω(n) w n ω(n) w n n = S(w,) < (log ) (2) for > 0. To bound the cardinality of B 3,2 (), note that each n B 3,2 () admits a prime divisor p such that ω(p ±) w. Fiing such a p, the number of such n does not eceed /p. Summing up over all such p we have, again in light of (), #B 3,2 () ω(p±) w p p ω(p+) w p+ + 2 p + + ω(p ) w p p (2S(w, + ) + S(w,)) < 3S(w,) + 2 (log ) (3) 5

6 for > 0. To bound the cardinality of B 3,3 (), note that for each n B 3,3 () there eist some prime r with ω(r ± ) w and some prime p n such that p ± (mod r). Given such r and p, the number of such n does not eceed /p. Summing up over all choices of r and p given above we get, again using (), for > 0. #B 3,3 () ω(r±) w r = (log log ) (log log ) p ± (mod r) p ω(r±) w r p ω(r ) w r r = (log log ) r + ω(r+) w r+ (log log )(S(w,) + 3S(w, + )) 4(log log )S(w,) + O(log log ) Hence, using estimates (2) to (4), we get We now let #B 3 () #B 3, () + #B 3,2 () + #B 3,3 () < ω(r±) w r 3 r + φ(r) (log log ) (log ) (4) (log ) 0 for > 0. (5) B 4 () = {n n (B () B 2 ()) and p 2 φ(n) for some p > z}. Let n B 4 () and let p 2 φ(n) for some prime p. Then it is not possible that p 2 n (because n B 2 ()), nor is it possible that p 2 q for some prime factor q of n (again because n B 2 ()). Thus, there must eist distinct primes q and r dividing n such that q (mod p) and r (mod p). Fiing such p, q and r, the number of acceptable values of such n does not eceed /(qr). Summing up over all the possible values of p, q and r we arrive at #B 4 () z p z p q (mod p) r (mod p) q<r, qr (log log )2 (log ) 3, qr z p 2 (log log ) 2 (p ) 2 (log log ) 2 6 q (mod p) q z p p 2 q 2

7 where we used estimates (8) and (5). Hence, We now let #B 4 () < (log ) 0 for all > 0. (6) τ = ep ( (log log ) 2) and let B 5 () stand for the set of n which are multiples of a prime p for which either p or p + has a divisor d > τ with P(d) < z. Fi such a pair d and p. Then the number of n divisible by p is at most /p. This shows that #B 5 () P(d)<z τ<d P(d)<z τ<d p ± (mod d) p p log log P(d)<z τ<d d. (7) It follows easily by partial summation from the estimates (3) for Ψ(,v), that if we write v = log τ/ log z, then S = d log ep(( + o())v log v). Since v = (log log )/26, we get that and therefore that v log v = (/26 + o())(log log )(log log log ) S for all > 0, which together with estimate (7) gives Thus, setting log ep(( + o())v log v) < (8) (log ) #B 5 () < (log ) 0 for > 0. (9) B() = we get, from estimates (4), (0), (5), (6) and (9) that 5 B i (), (20) i= #B() 5 i= #B i () (log ) 0 for all > 0. (2) 3 The Proof of Theorem We find it convenient to prove a stronger theorem. 7

8 Theorem 2. Let a and b be any fied positive integers. Setting A a,b = {n σ(aφ(n)) = bσ(n)}, then the estimate holds for all 3. #A a,b () a,b (log ) 2 Proof. Let be large and let B() be as in (20). We assume that n is a positive integer not in B(). We let A () be the set of n A a,b ()\B() for which (P(n) )/2 is not prime but such that there eists a prime number r > z and another prime number q P(n) for which r gcd(p(n) +,q + ). To count the number of such positive integers n, let r and q be fied primes such that r q +, and let P be a prime such that q P and r P +. The number of positive integers n such that P(n) = P does not eceed /P. Note that the congruences P (mod r) and P (mod q) are equivalent to P a q,r (mod qr), where a q,r is the smallest positive integer m satisfying m (mod r) and m (mod q). We distinguish two instances: Case : qr < P. Let A () be the set of such integers n A (). Then #A () z<r q (mod r) q log log z<r log log z<r P a q,r (mod qr) qr<p q (mod r) q r (log log ) 2 z<r (log log ) 2 z<r q (mod r) q rφ(r) where in the above inequalities we used estimates (8) and (5). Case 2: qr P. P φ(qr) q (log log )2, (22) r2 (log ) Let A () be the set of such integers n A (). Here we write n = Pm. Note that P > P(m) because y > z for large and n B () B 2 (). Furthermore, since r q +, we may write q = rl. Since q P, we may write P = sq + = s(rl )+ = srl+ s. Since r P +, we get that s (mod r), and therefore that s 2 (mod r). Hence, there eists a nonnegative integer λ such that s = λr + 2. If λ = 0, then s = 2 leading to 8

9 P = 2q +, which is impossible. Thus, λ > 0. Let us fi the value of λ as well as that of r. Then rl = q and (λr 2 + 2r)l (λr + ) = P (23) are two linear forms in the variable l which are simultaneously primes. Note that P /m and since P lr(λr + 2), we get that l /(mr(λr + 2)). In particular, mr(λr + 2). Recall that a typical consequence of Brun s sieve (see for eample Theorem 2.3 in [3]), is that if L (m) = Am + B and L 2 (m) = Cm + D are linear forms in m with integer coefficients such that AD BC 0 and if we write E for the product of all primes p dividing ABCD(AD BC), then the number of positive integers m y such that L (m) and L 2 (m) are simultaneously primes is Ky ( E ) 2 (log y) 2 φ(e) for some absolute constant K. Applying this result for our linear forms in l shown at (23) for which A = r, B =, C = λr 2 + 2r and D = (λr + ), we get that the number of acceptable values for l does not eceed K mr(λr + 2) (log(/(mr(λr + 2))) 2 (log log )2 mr(λr + 2), ( ) 2 (λr + 2)(λr + )r φ((λr + 2)(λr + )r) where for the rightmost inequality we used again the minimal order of the Euler function on the interval [,]. Here, K is some absolute constant. Summing up over all possible values of λ, r and m, we get #A () (log log ) 2 mr(λr + 2) z<r λ m ( ) ( < (log log ) 2 r 2 where we used again estimate (5). z<r (log log )2 (log ) 2 (log ) 3 = From estimates (22) and (24), we get #A () #A () + A () < λ ) ( λ m ) m (log log )2 (log ), (24) (log ) 0 for all > 0. (25) Now let A 2 () be the set of those n A a,b ()\(A () B()) and such that (P(n) )/2 is not prime. With n A 2 (), we get that n = Pm, where P > P(m) for > 0 because 9

10 y > z for large and n B () B 2 (). Then bσ(n) = bσ(m)(p + ). Let d be the largest divisor of P + such that P(d ) z. Then P + = d l, and bσ(n) = bσ(m)d l. Furthermore, φ(n) = φ(m)(p ), so that aφ(n) = aφ(m)(p ). Let d be the largest divisor of P which is z-smooth; that is, with P(d) z. Then P = dl. Note that l is squarefree (because n B 2 ()) and l and φ(m) are coprime (for if not, there would eist a prime r > z such that r l and r φ(m), so that r 2 φ(n), which is impossible because n B 4 ()). Since z > a for > 0, we get that aφ(m)d and l are coprime, and therefore that σ(φ(n)) = σ(aφ(m)d)σ(l). We thus get the equation σ(aφ(m)d)σ(l) = bσ(m)d l. Now note that l and σ(l) are coprime. Indeed, if not, since l is squarefree, there would eist a prime factor r of l (necessarily eceeding z) dividing q + for some prime factor q of l, that is of P. But this is impossible because n A (). Thus, l σ(aφ(m)d). Note now that Pm, so that m /P /y. Furthermore, ma{d,d } τ because n B 5 (). Let us now fi m, d and d. Then l σ(aφ(m)d), and therefore the number of choices for l does not eceed τ(σ(aφ(m)d)), where τ(k) is the number of divisors of the positive integer k. The above argument shows that if we write M for the set of such acceptable values for m, then #A 2 () τ2 y ma{τ(σ(aφ(m)d)) m M, d τ, d τ}. (26) To get an upper bound on τ(σ(aφ(m)d)), we write aφ(m)d as aφ(m)d = AB, where A is squarefull, B is squarefree, and A and B are coprime. Clearly, so that Since B is squarefree, it is clear that σ(b) Furthermore, σ(aφ(m)d) = σ(ab) = σ(a)σ(b), τ(σ(aφ(m)d)) τ(σ(a))τ(σ(b)). q aφ(m)d (q + ). ω(aφ(m)d) ω(aφ(n)) ω(a) + ω(n) + p n ω(p ) w 2 + w + O(), (27) so that we have ω(aφ(m)d) 2w 2 if > 0. The above inequalities follow from the fact that n B 3 (). Also, for each one of the at most 2w 2 prime factors q of aφ(m)d, the number q+ has at most w prime factors itself and its squarefull part does not eceed z, again because 0

11 n B 3 (). In conclusion, σ(b) C, where C is a number with at most 2w 3 prime factors whose squarefull part does not eceed z 2w2. Thus, where ω(c) 2w 3 and ρ(c) z 2w2. Clearly, and since also we finally get that τ(σ(b)) τ(c) 2 ω(c) τ(ρ(c)), τ(ρ(c)) ρ(c) z w2 = ep(w 2 log z) = ep(o(log log ) 3 ), 2 ω(c) 2 2w3 = ep(o(log log ) 3 ), τ(σ(b)) = ep(o(log log ) 3 ). (28) We now deal with σ(a). We first note that P(A) z for > 0. Indeed, if q > z divides A, then q 2 aφ(n). If > 0, then z > a, in which case the above divisibility relation forces q 2 φ(n) which is not possible because n B 3 (). By looking at the multiplicities of the prime factors appearing in A, we easily see that A ρ(a)ρ(d) ρ(p ) p m q aφ(m)d q z q ω(aφ(m)d) As we have seen at estimate (27), ω(aφ(m)d) 2w 2, in which case the above relation shows that A ρ(a)z ω(aφ(m)d)+ω(aφ(m)d)2 z 4w4 +2w 2 = ep(o(log log ) 5 ). But since σ(a) < A 2 and τ(σ(a)) σ(a) A 2, we get that which together with estimate (28) gives τ(σ(a)) = ep(o(log log ) 5 ), (29) τ(σ(aφ(m)d)) τ(σ(a))τ(σ(b)) = ep(o(log log ) 5 ). Returning to estimate (26), we get that ( #A 2 () τ2 y ep(o(log log )5 ) = ep log ) log log + O((log log )5 ) < for all > (log ) 0 0. (30) Thus, writing C() = A a,b ()\(A () A 2 () B()), we get that ( A a,b () = #C() + O (log ) 0. ). (3)

12 Moreover, if n C(), then n = Pm, with P > P(m) for > 0, and (P )/2 is a prime. Hence, P = 2q, where q is a Sophie Germain prime. Since also P /m, it follows by Brun s method that the number of such values for P is m(log /m) 2 (log log )2 = m(log y) 2 m(log ). 2 In the above inequalities we used the fact that /m P y. Summing up over all m, we get #C() (log log ) 2 (log log )2. (32) m(log ) 2 log m In particular, #A a,b () (log log )2. (33) log This is weaker than the bound claimed by our Theorem 2. However, it implies, by partial summation, that n + 2 t d(#a a,b(t)) n A a,b () = + #A a,b(t) t ( t= + O t=2 t(log log t) 2 dt 2 t 2 log t = O((log log ) 3 ). (34) To get some improvement, we return to C(). Let n C() and write it as n = Pm, where P > P(m). Write also P = 2q +. Then φ(n) = 2φ(m)q. Moreover, q > (y )/2 > z for > 0 so that q does not divide aφ(m) (because q > a and n B 3 ()). Hence, σ(aφ(n)) = σ(2aφ(m))(q + ). On the other hand, bσ(n) = bσ(m)(p + ) = 2bσ(m)(q + ). Thus, the equation σ(aφ(n)) = σ(bn) forces σ(2aφ(m)) = 2bσ(m), implying that m A 2a,2b (/P) A 2a,2b (/y). The argument which leads to estimate (32) now provides the better estimate In particular, we get #C() m A 2a,2b () #A a,b () (log log ) 2 m(log ) 2 (log log )5 (log ) 2. (log log )5 (log ) 2. (35) This is still somewhat weaker than what Theorem 2 claims. However, it implies that the sum of the reciprocals of the numbers in A a,b is convergent. In fact, by partial summation, ) 2

13 it follows, as in estimates (34), that n A a,b n y n y = #A a,b(t) t t d(#a a,b(t)) t= t=y ( + O y ) t(log log t) 5 t 2 (log t) dt 2 (log log y)5 + (log y) 2 y t(log t) 3/2dt (log y) /2. (36) We now take another look at C(). Let again n C() and write n = Pm. Fiing m and using the fact that P /m is a prime such that (P )/2 is also a prime, we get that the number of choices for P is m(log(/m)) 2. Hence, #C() m A 2a,2b (/y) m(log(/m)) 2. We now split the above sum at m = /2 and use estimate (36) to get #C() m A 2a,2b ( /2 ) (log /2 ) 2 m(log(/m)) 2 + m A 2a,2b m + (log y) 2 m A 2a,2b /2 m /y m A 2a,2b m /2 m(log(/m)) 2 (log ) + 2 (log y) = (log log )5/2 + 5/2 (log ) 2 (log ) 5/2 (log ) 2, (37) which together with estimate (3) leads to the desired conclusion of Theorem 2. m 4 Acknowledgments This work was done in April of 2006, while the second author was in residence at the Centre de recherches mathématiques of the Université de Montréal for the thematic year Analysis and Number Theory. This author thanks the organizers for the opportunity of participating in this program. He was also supported in part by grants SEP-CONACyT 46755, PAPIIT IN04505 and a Guggenheim Fellowship. The first author was supported in part by a grant from NSERC. 3

14 References [] E. R. Canfield, P. Erdős and C. Pomerance, On a problem of Oppenheim concerning Factorisatio Numerorum, J. Number Theory 7 (983), 28. [2] L. E. Dickson, A new etension of Dirichlet s theorem on prime numbers, Messenger of Math. 33 (904), [3] H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 974. [4] G. H. Hardy and J. E. Littlewood, Some problems on partitio numerorum III. On the epression of a number as a sum of primes, Acta Math. 44 (923), 70. [5] A. Hildebrand, On the number of positive integers and free of prime factors > y, J. Number Theory 22 (986), [6] A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. de Théorie des Nombres de Bordeau 5 (983), [7] A. Ivić, The Riemann Zeta-Function, Theory and Applications, Dover Publications, [8] A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (958), Erratum, Acta Arith. 5 (959), 259. [9] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Univ. Press, Mathematics Subject Classification: Primary A25; Secondary N37. Keywords: Euler function, sum of divisors. (Concerned with sequences A03363 and A37733.) Received June ; revised version received February Published in Journal of Integer Sequences, February Return to Journal of Integer Sequences home page. 4

On the largest prime factor of the Mersenne numbers

On the largest prime factor of the Mersenne numbers On the largest prime factor of the Mersenne numbers Kevin Ford Department of Mathematics The University of Illinois at Urbana-Champaign Urbana Champaign, IL 61801, USA ford@math.uiuc.edu Florian Luca Instituto

More information

On the number-theoretic functions ν(n) and Ω(n)

On the number-theoretic functions ν(n) and Ω(n) ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,

More information

CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

More information

On multiples with the same digit sum

On multiples with the same digit sum On multiples with the same digit sum Florian Luca Pantelimon Stănică March 9, 2008 Abstract Let g > be an integer and s g (m) be the sum of digits in base g of the positive integer m. In this paper, we

More information

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM Acta Math. Univ. Comenianae Vol. LXXXI, (01), pp. 03 09 03 ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM A. DUBICKAS and A. NOVIKAS Abstract. Let E(4) be the set of positive integers expressible

More information

Concentration of points on two and three dimensional modular hyperbolas and applications

Concentration of points on two and three dimensional modular hyperbolas and applications Concentration of points on two and three dimensional modular hyperbolas and applications J. Cilleruelo Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas Universidad

More information

Primality - Factorization

Primality - Factorization Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.

More information

On the greatest and least prime factors of n! + 1, II

On the greatest and least prime factors of n! + 1, II Publ. Math. Debrecen Manuscript May 7, 004 On the greatest and least prime factors of n! + 1, II By C.L. Stewart In memory of Béla Brindza Abstract. Let ε be a positive real number. We prove that 145 1

More information

On the largest prime factor of x 2 1

On the largest prime factor of x 2 1 On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical

More information

Uniform Distribution of the Fractional Part of the Average Prime Divisor

Uniform Distribution of the Fractional Part of the Average Prime Divisor Uniform Distribution of the Fractional Part of the Average Prime Divisor William D. Banks Department of Mathematics, University of Missouri Columbia, MO 652 USA bbanks@math.missouri.edu Moubariz Z. Garaev

More information

The sum of digits of polynomial values in arithmetic progressions

The sum of digits of polynomial values in arithmetic progressions The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr

More information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

Counting Primes whose Sum of Digits is Prime

Counting Primes whose Sum of Digits is Prime 2 3 47 6 23 Journal of Integer Sequences, Vol. 5 (202), Article 2.2.2 Counting Primes whose Sum of Digits is Prime Glyn Harman Department of Mathematics Royal Holloway, University of London Egham Surrey

More information

Homework until Test #2

Homework until Test #2 MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

More information

Prime and Composite Terms in Sloane s Sequence A056542

Prime and Composite Terms in Sloane s Sequence A056542 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.3 Prime and Composite Terms in Sloane s Sequence A056542 Tom Müller Institute for Cusanus-Research University and Theological

More information

ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS

ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract If n is a positive integer, write F n for the nth Fibonacci number, and ω(n) for the number

More information

Arrangements of Stars on the American Flag

Arrangements of Stars on the American Flag Arrangements of Stars on the American Flag Dimitris Koukoulopoulos and Johann Thiel Abstract In this article, we examine the existence of nice arrangements of stars on the American flag. We show that despite

More information

On the number of prime factors of an odd perfect number

On the number of prime factors of an odd perfect number On the number of prime factors of an odd perfect number Pascal Ochem CNRS, LIRMM, Université Montpellier 2 161 rue Ada, 34095 Montpellier Cedex 5, France ochem@lirmm.fr Michaël Rao CNRS, LIP, ENS Lyon

More information

The Pólya Vinogradov inequality

The Pólya Vinogradov inequality Illinois Number Theory Conference in honor of Harold Diamond at 70 The Pólya Vinogradov inequality Carl Pomerance, Dartmouth College Johann Peter Gustav Lejeune Dirichlet, quite the character... 1 What

More information

The cyclotomic polynomials

The cyclotomic polynomials The cyclotomic polynomials Notes by G.J.O. Jameson 1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(s + t) = e(s)e(t) for all s, t. e( 1 ) =

More information

FACTORING SPARSE POLYNOMIALS

FACTORING SPARSE POLYNOMIALS FACTORING SPARSE POLYNOMIALS Theorem 1 (Schinzel): Let r be a positive integer, and fix non-zero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S

More information

On the Union of Arithmetic Progressions

On the Union of Arithmetic Progressions On the Union of Arithmetic Progressions Shoni Gilboa Rom Pinchasi August, 04 Abstract We show that for any integer n and real ɛ > 0, the union of n arithmetic progressions with pairwise distinct differences,

More information

Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

More information

Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients

Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca

More information

PRIME FACTORS OF CONSECUTIVE INTEGERS

PRIME FACTORS OF CONSECUTIVE INTEGERS PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

Characterizing the Sum of Two Cubes

Characterizing the Sum of Two Cubes 1 3 47 6 3 11 Journal of Integer Sequences, Vol. 6 (003), Article 03.4.6 Characterizing the Sum of Two Cubes Kevin A. Broughan University of Waikato Hamilton 001 New Zealand kab@waikato.ac.nz Abstract

More information

Prime Numbers. Chapter Primes and Composites

Prime Numbers. Chapter Primes and Composites Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

More information

Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter 2 Limits Functions and Sequences sequence sequence Example Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

More information

Solutions to Practice Problems

Solutions to Practice Problems Solutions to Practice Problems March 205. Given n = pq and φ(n = (p (q, we find p and q as the roots of the quadratic equation (x p(x q = x 2 (n φ(n + x + n = 0. The roots are p, q = 2[ n φ(n+ ± (n φ(n+2

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005

Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005 Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005 A.J. Hildebrand Department of Mathematics University of Illinois http://www.math.uiuc.edu/~hildebr/ant Version 2013.01.07 18 Chapter

More information

FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS

FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS International Electronic Journal of Algebra Volume 6 (2009) 95-106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009

More information

Note on some explicit formulae for twin prime counting function

Note on some explicit formulae for twin prime counting function Notes on Number Theory and Discrete Mathematics Vol. 9, 03, No., 43 48 Note on some explicit formulae for twin prime counting function Mladen Vassilev-Missana 5 V. Hugo Str., 4 Sofia, Bulgaria e-mail:

More information

Gambling Systems and Multiplication-Invariant Measures

Gambling Systems and Multiplication-Invariant Measures Gambling Systems and Multiplication-Invariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously

More information

CONTINUED FRACTIONS AND RSA WITH SMALL SECRET EXPONENT

CONTINUED FRACTIONS AND RSA WITH SMALL SECRET EXPONENT CONTINUED FRACTIONS AND RSA WITH SMALL SECRET EXPONENT ANDREJ DUJELLA Abstract Extending the classical Legendre s result, we describe all solutions of the inequality α a/b < c/b in terms of convergents

More information

Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

More information

1.3 Induction and Other Proof Techniques

1.3 Induction and Other Proof Techniques 4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

More information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

More information

Problem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB

Problem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB Problem Set 5 1. (a) Four-digit number S = aabb is a square. Find it; (hint: 11 is a factor of S) (b) If n is a sum of two square, so is 2n. (Frank) Solution: (a) Since (A + B) (A + B) = 0, and 11 0, 11

More information

Power Series. Chapter Introduction

Power Series. Chapter Introduction Chapter 10 Power Series In discussing power series it is good to recall a nursery rhyme: There was a little girl Who had a little curl Right in the middle of her forehead When she was good She was very,

More information

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public

More information

Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

More information

4. Number Theory (Part 2)

4. Number Theory (Part 2) 4. Number Theory (Part 2) Terence Sim Mathematics is the queen of the sciences and number theory is the queen of mathematics. Reading Sections 4.8, 5.2 5.4 of Epp. Carl Friedrich Gauss, 1777 1855 4.3.

More information

MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS

MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS MATH 16300-33: HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS 25-1 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013 FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

More information

EULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.

EULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p. EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If

More information

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

More information

ON CERTAIN SUMS RELATED TO MULTIPLE DIVISIBILITY BY THE LARGEST PRIME FACTOR

ON CERTAIN SUMS RELATED TO MULTIPLE DIVISIBILITY BY THE LARGEST PRIME FACTOR Ann. Sci. Math. Québec 29 2005, no. 2, 3 45. ON CERTAIN SUMS RELATED TO MULTIPLE DIVISIBILITY BY THE LARGEST PRIME FACTOR WILLIAM D. BANKS, FLORIAN LUCA AND IGOR E. SHPARLINSKI RÉSUMÉ. Pour un nombre entier

More information

Solving a family of quartic Thue inequalities using continued fractions

Solving a family of quartic Thue inequalities using continued fractions Solving a family of quartic Thue inequalities using continued fractions Andrej Dujella, Bernadin Ibrahimpašić and Borka Jadrijević Abstract In this paper we find all primitive solutions of the Thue inequality

More information

Continued Fractions. Darren C. Collins

Continued Fractions. Darren C. Collins Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

More information

Is n a Prime Number? Manindra Agrawal. March 27, 2006, Delft. IIT Kanpur

Is n a Prime Number? Manindra Agrawal. March 27, 2006, Delft. IIT Kanpur Is n a Prime Number? Manindra Agrawal IIT Kanpur March 27, 2006, Delft Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 1 / 47 Overview 1 The Problem 2 Two Simple, and Slow, Methods

More information

Arrangements of Stars on the American Flag

Arrangements of Stars on the American Flag Arrangements of Stars on the American Flag Dimitris Koukoulopoulos and Johann Thiel Abstract. In this article, we examine the existence of nice arrangements of stars on the American flag. We show that

More information

Practice Problems for First Test

Practice Problems for First Test Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

More information

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

More information

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

About the inverse football pool problem for 9 games 1

About the inverse football pool problem for 9 games 1 Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

More information

PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

More information

AMBIGUOUS CLASSES IN QUADRATIC FIELDS

AMBIGUOUS CLASSES IN QUADRATIC FIELDS MATHEMATICS OF COMPUTATION VOLUME, NUMBER 0 JULY 99, PAGES -0 AMBIGUOUS CLASSES IN QUADRATIC FIELDS R. A. MOLLIN Dedicated to the memory ofd. H. Lehmer Abstract. We provide sufficient conditions for the

More information

RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER

RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER Volume 9 (008), Issue 3, Article 89, pp. RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER MARK B. VILLARINO DEPTO. DE MATEMÁTICA, UNIVERSIDAD DE COSTA RICA, 060 SAN JOSÉ,

More information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

b) Find smallest a > 0 such that 2 a 1 (mod 341). Solution: a) Use succesive squarings. We have 85 =

b) Find smallest a > 0 such that 2 a 1 (mod 341). Solution: a) Use succesive squarings. We have 85 = Problem 1. Prove that a b (mod c) if and only if a and b give the same remainders upon division by c. Solution: Let r a, r b be the remainders of a, b upon division by c respectively. Thus a r a (mod c)

More information

Students in their first advanced mathematics classes are often surprised

Students in their first advanced mathematics classes are often surprised CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

More information

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction. MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

More information

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear: MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

More information

ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

More information

Recent Breakthrough in Primality Testing

Recent Breakthrough in Primality Testing Nonlinear Analysis: Modelling and Control, 2004, Vol. 9, No. 2, 171 184 Recent Breakthrough in Primality Testing R. Šleževičienė, J. Steuding, S. Turskienė Department of Computer Science, Faculty of Physics

More information

MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

More information

Is n a prime number? Nitin Saxena. Turku, May Centrum voor Wiskunde en Informatica Amsterdam

Is n a prime number? Nitin Saxena. Turku, May Centrum voor Wiskunde en Informatica Amsterdam Is n a prime number? Nitin Saxena Centrum voor Wiskunde en Informatica Amsterdam Turku, May 2007 Nitin Saxena (CWI, Amsterdam) Is n a prime number? Turku, May 2007 1 / 36 Outline 1 The Problem 2 The High

More information

arxiv:math/ v2 [math.nt] 10 Nov 2007

arxiv:math/ v2 [math.nt] 10 Nov 2007 Non-trivial solutions to a linear equation in integers arxiv:math/0703767v2 [math.nt] 10 Nov 2007 Boris Bukh Abstract For k 3 let A [1, N] be a set not containing a solution to a 1 x 1 + + a k x k = a

More information

On Generalized Fermat Numbers 3 2n +1

On Generalized Fermat Numbers 3 2n +1 Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of Basic

More information

Algebraic and Transcendental Numbers

Algebraic and Transcendental Numbers Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)

More information

SOME TYPES OF CONVERGENCE OF SEQUENCES OF REAL VALUED FUNCTIONS

SOME TYPES OF CONVERGENCE OF SEQUENCES OF REAL VALUED FUNCTIONS Real Analysis Exchange Vol. 28(2), 2002/2003, pp. 43 59 Ruchi Das, Department of Mathematics, Faculty of Science, The M.S. University Of Baroda, Vadodara - 390002, India. email: ruchid99@yahoo.com Nikolaos

More information

Powers of Two in Generalized Fibonacci Sequences

Powers of Two in Generalized Fibonacci Sequences Revista Colombiana de Matemáticas Volumen 462012)1, páginas 67-79 Powers of Two in Generalized Fibonacci Sequences Potencias de dos en sucesiones generalizadas de Fibonacci Jhon J. Bravo 1,a,B, Florian

More information

k, then n = p2α 1 1 pα k

k, then n = p2α 1 1 pα k Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

More information

p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6) .9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

More information

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION

THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION ERICA CHAN DECEMBER 2, 2006 Abstract. The function sin is very important in mathematics and has many applications. In addition to its series epansion, it

More information

Double Sequences and Double Series

Double Sequences and Double Series Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions,

More information

8 Divisibility and prime numbers

8 Divisibility and prime numbers 8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

More information

10 k + pm pm. 10 n p q = 2n 5 n p 2 a 5 b q = p

10 k + pm pm. 10 n p q = 2n 5 n p 2 a 5 b q = p Week 7 Summary Lecture 13 Suppose that p and q are integers with gcd(p, q) = 1 (so that the fraction p/q is in its lowest terms) and 0 < p < q (so that 0 < p/q < 1), and suppose that q is not divisible

More information

Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression

On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression Carrie E. Finch and N. Saradha Abstract We prove the irreducibility of ceratin polynomials

More information

STRUCTURE AND RANDOMNESS IN THE PRIME NUMBERS. 1. Introduction. The prime numbers 2, 3, 5, 7,... are one of the oldest topics studied in mathematics.

STRUCTURE AND RANDOMNESS IN THE PRIME NUMBERS. 1. Introduction. The prime numbers 2, 3, 5, 7,... are one of the oldest topics studied in mathematics. STRUCTURE AND RANDOMNESS IN THE PRIME NUMBERS TERENCE TAO Abstract. A quick tour through some topics in analytic prime number theory.. Introduction The prime numbers 2, 3, 5, 7,... are one of the oldest

More information

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b. Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

More information

8.7 Mathematical Induction

8.7 Mathematical Induction 8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

More information

MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

More information

ARTICLE IN PRESS Nonlinear Analysis ( )

ARTICLE IN PRESS Nonlinear Analysis ( ) Nonlinear Analysis ) Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A question concerning a polynomial inequality and an answer M.A. Qazi a,,

More information

Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

More information

The Prime Numbers, Part 2

The Prime Numbers, Part 2 The Prime Numbers, Part 2 The fundamental theorem of arithmetic says that every integer greater than 1 can be written as a product of primes, and furthermore there is only one way to do it ecept for the

More information

arxiv:1112.0829v1 [math.pr] 5 Dec 2011

arxiv:1112.0829v1 [math.pr] 5 Dec 2011 How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

More information

CHAPTER 2. Inequalities

CHAPTER 2. Inequalities CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

More information