Proof of a supercongruence conjectured by Sun through a $${\varvec{q}}$$ q -microscope
- PDF / 228,792 Bytes
- 7 Pages / 510.236 x 680.315 pts Page_size
- 93 Downloads / 153 Views
Proof of a supercongruence conjectured by Sun through a q-microscope VICTOR J W GUO School of Mathematics and Statistics, Huaiyin Normal University, Huai’an 223300, Jiangsu, People’s Republic of China E-mail: [email protected]
MS received 18 December 2019; accepted 29 March 2020 Abstract. In 2011, Sun (Sci. China Math. 54 (2011) 2509–2535) made the following conjecture: for any odd prime p and odd integer m, 2k (m−1)/2 2k (mp−1)/2 2 1 k k ≡ 0 (mod p 2 ). − m−1 k 2 p 8 8k m (m−1)/2
k=0
k=0
By applying the, creative microscoping, method introduced by Guo and Zudilin (Adv. Math. 346 (2019) 329–35), we confirm the above conjecture of Sun. Keywords. Cyclotomic polynomial; q-binomial coefficient; q-congruence; super congruence; creative microscoping. Mathematics Subject Classification.
11B65, 11A07, 33D15.
1. Introduction During the past decade, congruences and supercongruences have been studied by quite a few authors. In 2011, Sun [14, Equation (1.6)] proved that, for any odd prime p, ( p−1)/2 2k k 8k k=0
≡
−2 p 2 2 + E p−3 (mod p 3 ), p p 4
where mn denotes the Jacobi symbol and E n is the nth Euler number. Later, Sun [15, Equation (1.7)] further proved that ( pr −1)/2 2k k 8k k=0
≡
2 pr
(mod p 2 ).
© Indian Academy of Sciences 0123456789().: V,-vol
(1.1)
63
Page 2 of 7
Proc. Indian Acad. Sci. (Math. Sci.)
(2020) 130:63
Recently, Sun [16, Conjecture 4(ii)] proposed the following conjecture: for any odd prime p and odd integer m,
m2
1
m−1 (m−1)/2
2k (mp−1)/2 k=0
k 8k
(m−1)/2 2k 2 k ≡ 0 (mod p 2 ), − p 8k
(1.2)
k=0
which is clearly a generalization of (1.1). In the past few years, q-analogues of congruences and supercongruences have caught the interest of a lot of people (see [3–13,17]). In particular, Guo and Liu [8] gave the following q-analogue of (1.1): for odd n > 1, (n−1)/2 k=0
qk
2
(q; q 2 )k 2 ≡ (−q)(1−n )/8 (mod n (q)2 ). 4 4 (q ; q )k
(1.3)
Here and in what follows, (a; q)n = (1 − a)(1 − aq) · · · (1 − aq n−1 ), n = 0, 1, . . . , or n = ∞, is the q-shifted factorial, and n (q) is the n-th cyclotomic polynomial in q given by (q − ζ k ), n (q) = 1kn gcd(n,k)=1
where ζ is an nth primitive root of unity. Moreover, Gu and Guo [4] gave some different q-analogues of (1.1), such as (n−1)/2 k=0
(q; q 2 )k q 2k ≡ (q 2 ; q 2 )k (−q; q 2 )k
2 2 q 2(n+1)/4 (mod n (q)2 ), n
(1.4)
where x denotes the largest integer not exceeding x. In order to prove Sun’s conjecture (1.2), we need the following new q-analogue of (1.1). Theorem 1.1. Let n > 1 be an odd integer. Then (n−1)/2 k=0
(q; q 2 )k (−1; q 4 )k 2k q ≡ (−q; q 2 )k (q 4 ; q 4 )k
2 (mod n (q)2 ). n
(1.5)
n−1 and the q-binomial Recall that m the q-integer is defined by [n]q = 1 + q + · · · + q coefficient n q is defined as
⎧ (q; q)m
⎨ , if 0 n m, m = (q; q)n (q; q)m−n n q ⎩0, otherwise. On the basis of (1.5), we are able to give the following q-analogue of (1.2).
Proc. Indian Acad. Sci. (Math. Sci.)
(2020) 130:63
Page 3
Data Loading...
