How To Find Out If A Rie Is A Power Of 2 Or More (For A Ri)

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1 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) DONALD M. DAVIS AND ZHI-WEI SUN Abstract. We use ethods of cobinatorial nuber theory to rove that, for each n 2 and any rie, soe hootoy grou π i SUn)) contains an eleent of order n 1+ord n/!), where ord ) denotes the largest integer such that. 1. Introduction Let be a rie nuber. The hootoy -exonent of a toological sace X, denoted by ex X), is defined to be the largest e N {0, 1, 2,... } such that soe hootoy grou π i X) has an eleent of order e. This concet has been studied by various toologists cf. [12], [10], [15], [3], [4], [5], [14], [18], and [19]). The ost celebrated result about hootoy exonents roved by Cohen, Moore, and Neisendorfer in [3]) states that ex S 2n+1 ) n if 2. The secial unitary grou SUn) of degree n) is the sace of all n n unitary atrices the conugate transose of such a colex atrix equals its inverse) with deterinant one. See, e.g., [11,. 68].) It lays a central role in any areas of atheatics and hysics. The faous Bott Periodicity Theore [2]) describes π i SUn)) with i < 2n. In this aer, we rovide a strong and elegant lower bound for the hootoy -exonent of SUn). As in nuber theory, the integral art of a real nuber c is denoted by c. For a rie and an integer, the -adic order of is given by ord ) su{n N : n } whence ord 0) + ). Here is our ain result. Key words and hrases. hootoy grou, unitary grou, -adic order, binoial coefficient Matheatics Subect Classification: 55Q52, 57T20, 11A07, 11B65, 11S05. The second author is suorted by the National Science Fund for Distinguished Young Scholars no ) in P. R. China. 1

2 2 DONALD M. DAVIS AND ZHI-WEI SUN Theore 1.1. For any rie and n 2, 3,..., soe hootoy grou π i SUn)) contains an eleent of order n 1+ord n/!) ; i.e., we have the inequality ) n ex SUn)) n 1 + ord!. We discuss in Section 2 the extent to which Theore 1.1 ight be shar. Our reduction fro hootoy theory to nuber theory involves Stirling nubers of the second ind. For n, N with n + Z + {1, 2, 3,... }, the Stirling nuber Sn, ) of the second ind is the nuber of artitions of a set of cardinality n into nonety subsets; in addition, we define S0, 0) 1. We will use the following definition. Definition 1.2. Let be a rie. For, n Z + with n, we define e n, ) in n ord!s, )). In Sections 2 and 4 we rove the following standard result. Proosition 1.3. Let be a rie, and let n Z +. Then, for all n, we have ex SUn)) e n, ) unless 2 and n 0 od 2), in which case ex 2 SUn)) e 2 n, ) 1. Our innovation is to extend revious wor [16]) of the second author in cobinatorial nuber theory to rove the following result, which, together with Proosition 1.3, iediately ilies Theore 1.1 when or n is odd. In Section 4, we exlain the extra ingredient required to deduce Theore 1.1 fro 1.3 and 1.4 when 2 and n is even. Theore 1.4. Let be any rie and n be a ositive integer. i) For any, h, l, N, we have ) l l ord! 1) S h 1) + n 1, ) 0 { )} in l + 1), n 1 + ord!.

3 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) 3 ii) If we define N n 1 + n/ 1)), then ) e n, 1) L + n 1 ) n 1+ord n! for L N, N +1,.... In Section 3, we rove the following broad generalization of Theore 1.4, and in Section 2, we show that it ilies Theore 1.4. Theore 1.5. Let be a rie,, n N and r Z. Then for any olynoial fx) Z[x] we have ord r od ) n 1) f ) ) r n ord!. Here we adot the standard convention that n is 0 if is a negative integer. In Theore 5.1, we give a strengthened version of Theore 1.5, which we conecture to be otial in a certain sense. Our alication to toology uses the case r 0 of Theore 1.5; the ore technical Theore 5.1 yields no iroveent in this case. In [5], the first author used totally different, and uch ore colicated, ethods to rove that n n ex SUn)) n 1 + +, where is an odd rie and n is an integer greater than one. Since ord!) for every 0, 1, 2,... i1 i ) a well-nown fact in nuber theory), the inequality in Theore 1.1 can be restated as n ex SUn)) n 1 +, i2 i a nice iroveent of 1.6). 2. Outline of roof In this section we resent the deduction of Theore 1.1 fro Theore 1.5, which will then be roved in Section 3. We also resent soe coents regarding the extent to which Theore 1.1 is shar. Let be any rie. In [8], the first author and Mahowald defined the -riary) v 1 -eriodic hootoy grous v 1 1 π X; ) of a toological sace X and roved that if

4 4 DONALD M. DAVIS AND ZHI-WEI SUN X is a shere or coact Lie grou, such as SUn), each grou v 1 1 π i X; ) is a direct suand of soe actual hootoy grou π X). See also [7] for another exository account of v 1 -eriodic hootoy theory. In [6, 1.4] and [1, 1.1a], it was roved that if is odd, or if 2 and n is odd, then there is an isoorhis v 1 1 π 2 SUn); ) Z/ en,) Z 2.1) for all n, where e n, ) is as defined in 1.2 and we use Z/Z to denote the additive grou of residue classes odulo. Thus, unless 2 and n is even, for any integer n, we have ex SUn)) e n, ), establishing Proosition 1.3 in these cases. The situation when 2 and n is even is soewhat ore technical, and will be discussed in Section 4. Next we show that Theore 1.5 ilies Theore 1.4. Proof of Theore 1.4. i) By a well-nown roerty of Stirling nubers of the second ind cf. [13, ]),!Sh 1) + n 1, ) 0 1) h 1) +n 1 for any N. Thus l l 1)! 1) Sh 1) + n 1, ) Σ 1 + Σ 2, where Σ 1 and l 0 Σ 2 0 l 1) n 1+h 1) 0 od ) 0 od ) n 1+h 1) 1) l l 1) 1) n 1+h 1) 0 od ) 0 1) n 1 1 h 1)) l. Clearly ord Σ 1 ) n 1 + ord /!) by Theore 1.5, and ord Σ 2 ) l + 1) by Euler s theore in nuber theory. Therefore the first art of Theore 1.4 holds.

5 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) 5 ii) Observe that n N + 1 n 1) > 1) n ) i2 > n n ord i i2 i!. By art i) in the case l h 1 and L N, if n then ord!sn 1, )!S 1) L + n 1, ) ) n 1 + ord n ) Since Sn 1, ) 0 for n, we finally have as required. e n, 1) L + n 1) n 1 + ord n!. )! The following roosition, although not needed for our ain results, sheds ore light on the large exonents N and L which aear in Theore 1.4ii), and is useful in our subsequent exosition. Proosition 2.2. Let be a rie and let n > 1 be an integer. Then there exists an integer N 0 0, effectively coutable in ters of and n, such that e n, 1) L + n 1) has the sae value for all L N 0. Proof. For integers n and L 0, we write 1)!S 1) L +n 1, ) 1) 1)L +n 1 S +S,L+S,L, where S S,L S,L 0 1) n 1, 0 od ) 1) n 1 1)L 1), 0 od ) 1) 1)L +n 1. 0 od ) Note that both S,L and S,L are divisible by L+1. Assue that S n, S n+1,... are not all zero. This will be shown later.) Then L 0 in n ord S ) is finite. Let 0 n satisfy ord S 0 ) L 0. Whenever

6 6 DONALD M. DAVIS AND ZHI-WEI SUN L L 0, we have ord S + S,L + S,L) L 0 for every n, and equality is attained for 0. Thus, if L L 0 then e n, 1) L + n 1) in n ord!s 1) L + n 1, )) in n ord S + S,L + S,L) L 0. Although L 0 is finite, it ay not be effectively coutable. Instead of L 0 we use the -adic order N 0 of the first nonzero ter in the sequence S n, S n+1,.... This N 0 is coutable, also e n, 1) L + n 1) L 0 for all L N 0 since N 0 L 0. To colete the roof, we ust show that S is nonzero for soe n. First note that this is clearly true for 2 since then S is a su of negative ters. If is odd and S 0 for all n, then e n, 1) L +n 1) in n ord S,L +S,L) L + 1 for any L 0. By 2.1), this would ily that v 1 1 π SUn); ) has eleents of arbitrarily large -exonent. However, this is not true, for in [6, 5.8], it was shown that the v 1 -eriodic -exonent of SUn) does not exceed e : n 1)1+ 1) 1 + 1) 2 ) ; i.e., for this e, e v 1 1 π SUn); ) 0. In the reainder of this section and in Section 4, once a rie and an integer n > 1 is given, L will refer to any integer not saller than ax{n, N 0 } where N and N 0 are described in Theore 1.4ii) and the roof of Proosition 2.2 resectively. We now coent on the extent to which Theore 1.1 ight be shar. In Table 1, we resent, for 3 and a reresentative set of values of n, three nubers. The first, labelled ex 3 v 1 1 SUn)), is the largest value of e 3 n, ) over all values of n; thus it is the largest exonent of the 3-riary v 1 -eriodic hootoy grous of SUn). The second nuber in the table is the exonent of the v 1 -eriodic hootoy grou on which we have been focusing, which, at least in the range of this table, is equal to or ust slightly less than the axial exonent. The third nuber is the nice estiate for this exonent given by Theore 1.4. Note that, for ore than half of the values of n in the table, the largest grou v 1 1 π 2 SUn); 3) occurs when 2 3 L + n 1. In the worst case in the table, n 29, detailed Male calculations suggest that if 29 and 10 od 18), then e 3 29, ) in{ord ) + 12, 34}.

7 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) 7 Table 1. Coarison of exonents when 3 n ex 3 v1 1 SUn)) e 3 n, 2 3 L + n 1) n 1 + ord 3 n/3!) Shifts as by ) were already noted in [6,. 543]. Note also that for ore than half of the cases in the table, our estiate for e 3 n, 2 3 L + n 1) is shar, and it never isses by ore than 3. The big question for toologists, though, is whether the v 1 -eriodic -exonent agrees or alost agrees) with the actual hootoy -exonent. The fact that they agree for S 2n+1 when is an odd rie [3], [10]) leads the first author to conecture that they will agree for SUn) if 2, but we have no idea how to rove this. Theriault [18], [19]) has ade good rogress in roving that soe of the first author s lower bounds for -exonents of certain excetional Lie grous are shar.

8 8 DONALD M. DAVIS AND ZHI-WEI SUN 3. Proof of Theore 1.5 In this section, we rove Theore 1.5, which we have already shown to ily Theore 1.1. Lea 3.1. Let be any rie, and let, n N and r Z. Then ) ) n n n n ord ) 1) ord! +ord 1!. r od ) Proof. The equality is easy, for, ) n n/ 1 n ord! 1 i1 i n n/ ) n n + + ord i1 i!. When 0 or n < 1, the desired inequality is obvious. Now let > 0 and n/ 1 1. Observe that ord!) < i 1 i i1 i1 Thus 1) ord!) 1, and hence 1 n/ 1 1 n 1 ord!), 1 1 ϕ ) where ϕ is Euler s totient function. By a result of Weisan [21], n n ord ) 1) 1. ϕ ) r od ) 0 Weisan s roof is colicated, but an easy induction roof aeared in [16].) So we have the desired inequality. Now we restate Lea 2.1 of Sun [16], which will be used later. Lea 3.2. [16]) Let and n be ositive integers, and let fx) be a function fro Z to a field. Then, for any r Z, we have ) n n r 1) n 1 r f 1) 1 f, 0 rod ) where r r 1 + and fx) fx + 1) fx).

9 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) 9 Lea 3.3. Let, n Z + and r Z, and let fx) be a colex-valued function defined on Z. Then we have ) n r n r 1) n f f 1) r od ) r od ) n 1 n 1) i n 1 1) r f, i 0 i r od ) r od ) where r r + 1 and fx) fx + 1) fx). Proof. Let ζ be a riitive th root of unity. Clearly 1 1 ζ r)s s0 1 if r od ), 1 ζ r) s0 0 otherwise. 1 ζ r Thus n r 1) f r od ) ) n 1 1 n r ζ r)s 1) f 1 0 s0 1 s0 ζ rs c s, where ) n n r c s ζ s ) f. 0 Observe that ) n n r c s 1 ζ s ) 1) f 0 n n ) r 1 ζ s ) 1) f 0 0 n n n ) 1 ζ s ) n r 1) f 0 n n n ) )1 ζ s ) n r ) 1) f. 0 0 Alying Lea 3.2, we find that ) n r c s 1 ζ s ) n f n 1 n 1 ζ s ) n 1 1) 1 r f. 0 r od )

10 10 DONALD M. DAVIS AND ZHI-WEI SUN In view of the above, it suffices to note that ζ rs 1) 1 ζs ) i 1 ζ si r) i 1 s0 i0 This concludes the roof. s0 i r od ) 1) i. i With hel of Leas 3.1 and 3.3, we are able to rove the following equivalent version of Theore 1.5. Theore 3.4. Let be a rie, and let, l, n N. Then for any r Z we have l ) ) n r n ord 1) ord!. r od ) Proof. We use induction on l. In the case l 0, the desired result follows fro Lea 3.1. Now let l > 0 and assue the result for saller values of l. We use induction on n to rove the inequality in Theore 3.4. The case n 0 is trivial. So we now let n > 0 and assue that the inequality holds with saller values of n. Observe that l n r 1) r od ) ) n 1 n 1 r + 1) 1 r od ) l n 1 r 1) r od ) n 1 ) l r 1) 1). r 1 od ) In view of this, if does not divide n, then, by the induction hyothesis for n 1, we have l ) n r ord 1) r od ) ) ) n 1 n ord ord. ) l

11 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) 11 Below we let n and set n/. Case 1. r 0 od ). In this case, 1 n 1)! r ) l 1 r r od ) n/ l 1 n 1 r 1)! 1 r od ) r/ l 1 n r 1)! r od ) 1 n 1 r 1) 1) +1 n 1)/! r 1 od ) r/ l 1 n r 1). n/! r od ) Thus, by the induction hyothesis for l 1, 1 n r 1)! r od ) is a -integer i.e., its denoinator is relatively rie to ) and hence the desired ) l ) l 1 inequality follows. Case 2. r 0 od ). Note that 0 i r od ) i 1) i 0. Also, n n ord ) 1) ord! + ord 1!) r od ) by Lea 3.1. Thus, in view of Lea 3.3, it suffices to show that if 0 < < n then the -adic order of n σ i r od ) 1) i i r od ) n 1 1) r f is at least ord!), where r r + 1 and fx) x l. Let 0 < n 1 and write s + t, where s, t N and t <. Note that n 1 s and s t + 1 s 1.

12 12 DONALD M. DAVIS AND ZHI-WEI SUN Since fx) x+1) l x l l 1 l i0 i) x i, by Lea 3.1 and the induction hyothesis with resect to l, we have n ord σ ) ord + ord ) 1) i i i r od ) ) n 1 + ord 1) r f r od ) ) n ord + s + ord s!)) + ord s 1)!) ) ) n s ord + s + ord s!) ord i) + ord!) i0 ) ord ord + s ord s) + ord!). s + t s When t 0 i.e., s) we have the stronger inequality ) ord σ ) ord ord + s + ord!), s s because ord n ord r od ) ) 0 1 ord i0 ord n! n ) n 1 1) r f )) r ) 1) f by Lea 3.2) r ) i od ) ) n ord s)!) ) 1) r ) i ) l ) by the induction hyothesis with resect to n). Observe that ) ord s ) s s) i1 i i i s ) s) + i+1 i i i ) s s ord i1 i i i. s

13 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) 13 Thus, when t 0 we have ord σ ) s + ord!) ord!). then Define ord a/b) ord a) ord b) if a, b Z and a is not divisible by b. If t > 0 ) ord ord s + t s ) ord s+t s ord s) s + t For 0 < i <, clearly s)! s))! ) ord s + t)! s) t)! + ord 0<i<t s) i. s + i ord s) i) ord s + i) ord i) <. Therefore, when 0 < t < we have ) ord ord ord s)+ ord s+t) > ord s) s + t s and hence ord σ ) > s + ord!) ord!). This concludes the analysis of the second case. The roof of Theore 3.4 is now colete. Note that, in the roof of Theore 3.4, the technique used to handle the first case is of no use in the second case, and vice versa. Thus, the distinction of the two cases is iortant. Corollary 3.5. Let be a rie, and let, l, n N and r Z. Then we have ) ) n r)/ ord 1) n ord! ord l r od ) l!). Proof. Sily aly Theore 1.5 with fx) l! x l Z[x]. 4. Changes when 2 and n is even When 2 and n is even, the relationshi between v1 1 π 2 SUn); ) and e n, ) with n) is not so sile as in 2.1). As described in [1] and [9], there is a sectral 1, 2+1 sequence converging to v1 1 π SUn); ) and satisfying E2 SUn)) Z/ en,) Z. If or n is odd, the sectral sequence necessarily collases and v1 1 π 2 SUn); )

14 14 DONALD M. DAVIS AND ZHI-WEI SUN 1, 2+1 E 2. Here we begin abbreviating Er, SUn)) ust as Er,.) If 2 and n is even, there are two ways in which the corresonding suand of v1 1 π 2 SUn); 2) ay differ fro this. It is conceivable that there could be an extension in the sectral sequence, which would ae the exonent of the hootoy grou 1 larger than that of E 1, 2+1. However, as observed in [9, 6.21)], it is easily seen that this does not haen. 1, 2+1 4, 2+3 It is also conceivable that the differential d 3 : E3 E 3 Z/2Z Z/2Z could be nonzero, which would ae the exonent of v1 1 π 2 SUn); 2) equal to e 2 n, ) 1. This is the reason for the 1 at the end of Proosition 1.3. By [9, 1.6], if n 0 od 4) and 2 L 1, 2+1 4, n 1, then d 3 : E3 E3 ust be 0. Now suose n 2 od 4). If n 2, then n 1+ord 2 n/2!) 1 < ex 2 SUn)) since π 6 SU2)) Z/12Z [20]). Below we let n > 2, hence n/2 + 1 is even and not larger than n 1. As first noted in [1, 1.1] and restated in [9, 6.5], for 2 L + n 1, 1, 2+1 4, 2+3 d 3 : E3 E3 is nonzero if and only if e 2 n, 2 L + n 1) e 2 n 1, 2 L + n 1) + n 1. We show at the end of the section that e 2 n 1, 2 L + n 1) ord 2 n 1)!). 4.1) Thus, if the above d 3 is nonzero, then e 2 n, 2 L + n 1) n 1 + ord 2 n 1)!) and hence ex 2 SUn)) e 2 n, 2 L +n 1) 1 n 1+ord 2 n 1)!) 1 n 1+ord 2 n/2!), as claied in Theore 1.1. Proof of 4.1). Putting 2, L, l h 1 and n 1 in the first art of Theore 1.4, we get that ord 2 n 1)!Sn 1, n 1) n 1)!S2 L + n 1, n 1) ) ) n 1 n 1 + ord 2! n 1 > ord 2 n 1)!). 2 Therefore ord 2 n 1)!S2 L + n 1, n 1)) ord 2 n 1)!). On the other hand, by the second art of Theore 1.4, ord 2!S2 L + n 1, )) n 1 + ord 2 n/2!) for all n. So we have 4.1).

15 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) Strengthening and sharness of Theore 3.4 In this section, we give an exale illustrating the extent to which Theore 3.4 is shar when r 0, which is the situation that is used in our alication to toology. Then we show in Theore 5.1 that the lower bound in Theore 3.4 can soeties be increased slightly. We begin with a tyical exale of Theore 3.4. n 100. Then n/ 25 and ord n/!) 22. For l 25, set l ) n δl) ord od 4) Let 2, r 0 and The range l n/ 25 is that in which we feel Theore 3.4 to be very strong. See Rear 5.32).) Clearly δl) easures the aount by which the actual -adic order of the su in Theore 3.4 exceeds our bound for it. The values of δl) for 25 l 45 are given in order as 0, 0, 0, 0, 2, 3, 2, 4, 1, 1, 1, 1, 2, 2, 4, 1, 0, 0, 0, 0, 3. When r 0 and in any other situations, Theore 3.4 aears to be shar for infinitely any values of l. Before resenting our strengthening of Theore 3.4 we need soe notation. For a Z and Z +, we let {a} denote the least nonnegative residue of a odulo. Given a rie, for any a, b N we let τ a, b) reresent the nuber of carries occurring in the addition of a and b in base ; actually ) a + b a b a + b τ a, b) ord i i i a i1 as observed by E. Kuer. Here is our strengthening of Theore 3.4. The right hand side is the aount by which the bound in Theore 3.4 can be iroved. This aount does not exceed, by the definition of τ. In Table 2, we illustrate this aount when 3 and 2. Theore 5.1. Let be a rie, and let, l, n N. Then, for all r Z, we have l ) ) n r n ord 1) ord r od )! {r} + {n r} τ {r}, {n r} ) ord. {r}

16 16 DONALD M. DAVIS AND ZHI-WEI SUN Proof. We use induction on n. In the case n 0, whether r 0 od ) or not, the desired result holds trivially. Now let n > 0 and assue the corresonding result for n 1. τ {r}, {n r} ) > 0. Then neither r nor n r is divisible by. and Set Clearly R and thus R n/ n/! 1 n/! R n/ n/! 1 n/! r od ) r 1 od ) r R + R 1 n/! r od ) r od ) l n r 1) l n 1 r 1) 1). r od ) l n 1 r 1) 1 n 1) l r, l+1 n r 1). This is a -integer by Theore 3.4; therefore ord rr + R ). Let β ord n). We consider three cases. Suose that Case 1. β. In this case, n/!/n/ ) n 1)/! and hence R is a -integer by Theore 3.4. In view of the inequality ord rr + R ), we have ord R) ord r) τ {r}, {n r} ), where the last equality follows fro the definition of τ and the condition n 0 r od ). that Case 2. ord r) β <. Since n/ n 1)/, the definition of R ilies R n 1 n 1)/! r 1 od ) Alying the induction hyothesis, we find that l n 1 r 1) 1). ord R ) β τ {r 1}, {n 1 r 1)} ) τ {r 1}, {n r} ).

17 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) 17 Since {r} + {n r} n 0 od ) and {r} + {n r} {r} + {n r} {r} + {n r} 1 {r} {r} {r} 1 {r} + {n r} {r 1} + {n r}, {r} {r 1} we have Thus τ {r}, {n r} ) τ {r 1}, {n r} ) + β ord r). ord R ) ord r) + τ {r}, {n r} ). Clearly τ {r}, {n r} ) ord r) by the definition of τ, so we also have Therefore r od ) ord rr + R ) ord r) + τ {r}, {n r} ). ord R) ord rr) ord r) τ {r}, {n r} ). Case 3. β < in{, ord r)}. In this case, ord r) β < where r n r. Also, l n r 1) l n r n ) 1) n r od ) 1) l+n l n r 1). Thus, as in the second case, we have 1 ord R) ord n/! r od ) r od ) l ) n r 1) τ { r}, {n r} ) τ {r}, {n r} ). The induction roof of Theore 5.1 is now colete. The following conecture is based on extensive Male calculations. Conecture 5.2. Let be any rie. And let, l N, n, r Z, with n 2 1. Then equality in Theore 5.1 is attained if l n/ and r n r od l + ) 1) log n/ ).

18 18 DONALD M. DAVIS AND ZHI-WEI SUN Rear ) The conecture, if roved, would show that Theore 5.1 would be otial in the sense that it is shar for infinitely any values of l. 2) Note that the conecture only deals with equality when l n/. For saller values of l, our inequality is still true, but not so strong. In [17], we obtain a stronger inequality when l < n/. We close with a table showing the aount by which the bound in Theore 5.1 iroves on that of Theore 3.4. That is, we tabulate τ {r}, {n r} ) when 3 and 2. Table 2. Values of τ 3 {r} 9, {n r} 9 ) {r} {n} References [1] M. Bendersy and D. M. Davis, 2-riary v 1 -eriodic hootoy grous of SUn), Aer. J. Math ) [2] R. Bott, The stable hootoy of the classical grous, Annals of Math ) [3] F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double susension and exonents of the hootoy grous of sheres, Annals of Math ) [4], Exonents in hootoy theory, Annals of Math. Studies ) [5] D. M. Davis, Eleents of large order in π SUn)), Toology ) [6], v 1 -eriodic hootoy grous of SUn) at odd ries, Proc. London Math. Soc ) [7], Couting v 1 -eriodic hootoy grous of sheres and certain Lie grous, Handboo of Algebraic Toology, Elsevier, 1995,

19 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SUn) 19 [8] D. M. Davis and M. Mahowald, Soe rears on v 1 -eriodic hootoy grous, London Math. Soc. Lect. Notes ) [9] D. M. Davis and K. Potoca, 2-riary v 1 -eriodic hootoy grous of SUn) revisited, to aear in Foru Math, on-line version: htt:// dd1/sun2long.df. [10] B. Gray, On the shere of origin of infinite failies in the hootoy grous of sheres, Toology ) [11] D. Huseoller, Fibre Bundles, 2nd ed., Sringer, [12] I. M. Jaes, On the susension sequence, Annals of Math ) [13] J.H. van Lint and R. M. Wilson, A Course in Cobinatorics, 2nd ed., Cabridge Univ. Press, Cabridge, [14] J. A. Neisendorfer, A survey of Anic-Gray-Theriault constructions and alications to exonent theory of sheres and Moore saces, Conte. Math. Aer. Math. Soc ) [15] P. Selic, Odd-riary torsion in π S 3 ), Toology ) [16] Z. W. Sun, Polynoial extension of Flec s congruence, Acta Arith ) [17] Z. W. Sun and D. M. Davis, Cobinatorial congruences odulo rie owers, to aear in Trans. Aer. Math. Soc., on-line version: htt://arxiv.org/abs/ath.nt/ [18] S. D. Theriault, 2-riary exonent bounds for Lie grous of low ran, Canad. Math. Bull ) [19], The 5-riary hootoy exonent of the excetional Lie grou E 8, J. Math. Kyoto Univ ) [20] H. Toda, A toological roof of theores of Bott and Borel-Hirzebruch for hootoy grous of unitary grous, Me. Coll. Sci. Univ. Kyoto ) [21] C. S. Weisan, Soe congruences for binoial coefficients, Michigan Math. J ) Deartent of Matheatics, Lehigh University, Bethlehe, PA 18015, USA E-ail address: dd1@lehigh.edu Deartent of Matheatics, Naning University, Naning , Peole s Reublic of China E-ail address: zwsun@nu.edu.cn