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Open conjectures on congruences (0)

by Z W Sun
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Congruences concerning Legendre polynomials

by Zhi-hong Sun - PROC. AMER. MATH. SOC , 2011
"... Let p be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for k=0 k confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences. ..."
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Let p be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for k=0 k confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.
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... p2 ) if p ≡ 5, 7 (mod 8) and ∑p−1 k=0 ( 2k k )2( ) 3k k ≡ k 108 { 4x 2 − 2p (mod p 2 ) if 3 | p − 1 and so p = x 2 + 3y 2 , 0 (mod p 2 ) if 3 | p − 2. Recently the author’s twin brother Zhi-Wei Sun (=-=[13,14]-=-) made a lot of conjectures on supercongruences. In particular, he conjectured that for a prime p = 2, 7, ∑p−1 k=0 ( ) 3 p−1 2k ∑ ≡ k k=0 (4k)! 81k { 2 2 p 4x − 2p (mod p ) if ( ≡ k! 4 0 (mod p 2 ) i...

CONGRUENCES FOR CENTRAL BINOMIAL SUMS AND FINITE POLYLOGARITHMS

by Sandro Mattarei, Roberto Tauraso , 2011
"... ..."
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PRODUCTS AND SUMS DIVISIBLE BY CENTRAL BINOMIAL COEFFICIENTS

by Zhi-wei Sun , 2010
"... In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that ..."
Abstract - Cited by 9 (3 self) - Add to MetaCart
In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that

WOLSTENHOLME’S THEOREM: ITS GENERALIZATIONS AND EXTENSIONS IN THE LAST HUNDRED AND FIFTY YEARS (1862–2012)

by Romeo Mestrovic , 2011
"... ..."
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GENERALIZED LEGENDRE POLYNOMIALS AND RELATED CONGRUENCES MODULO p²

by Zhi-hong Sun , 2012
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...∑ k=0 ( 3k k )( 6k 3k ) 432k ≡ (−1 p ) (mod p2),(1.6) where (ap ) is the Legendre symbol. These congruences were later confirmed by Mortenson[M1-M3] via the Gross-Koblitz formula. Recently Zhi-Wei Sun=-=[Su1]-=- posed more conjectures concerning the following sums modulo p2: p−1∑ k=0 ( 2k k )2 16k xk, p−1∑ k=0 ( 2k k )( 3k k ) 27k xk, p−1∑ k=0 ( 2k k )( 4k 2k ) 64k xk, p−1∑ k=0 ( 3k k )( 6k 3k ) 432k xk. As ...

Congruences involving ...

by Zhi-hong Sun , 2013
"... Let p> 3 be a prime, and let m be an integer with p ∤ m. In this paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring’s theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning p−1 k=0 ..."
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Let p> 3 be a prime, and let m be an integer with p ∤ m. In this paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring’s theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning p−1 k=0
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...by Mortenson[M] and Zhi-Wei Sun[Su2]. Let Z be the set of integers, and for a prime p let Zp be the set of rational numbers whose denominator is coprime to p. Recently the author’s brother Zhi-Wei Sun=-=[Su1]-=- posed many conjectures for ∑p−1 k=0 ( 2k k )2(4k 2k ) m−k (mod p2), where p > 3 is a prime and m ∈ Z with p ∤ m. For example, he conjectured (see [Su1, Conjecture A3]) (1.1) p−1∑ k=0 ( 2k k )2(4k 2k ...

ON DIVISIBILITY CONCERNING BINOMIAL COEFFICIENTS

by Zhi-wei Sun , 2010
"... Let k,l and n be positive integers. We mainly show that (ln+1) ∣ ( kn+ln k, kn kn ..."
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Let k,l and n be positive integers. We mainly show that (ln+1) ∣ ( kn+ln k, kn kn

3 /m k

by Zhi-hong Sun
"... Abstract. Let p> 3 be a prime, and let m be an integer with p ∤ m. In the paper we p−1 k=0 ..."
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Abstract. Let p> 3 be a prime, and let m be an integer with p ∤ m. In the paper we p−1 k=0
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...p ) p−1 ∑ x=0 ( x 3 − 3 2 (3t + 5)x + 9t + 7) (mod p). p From (1.8) we deduce (1.7) immediately. As consequences of (1.8), we determine P p [ 4 ) (mod p) and use them to solve Z.W. Sun’s conjectures (=-=[Su1]-=-) on 7 P p [ 4 ](− 9 ), P [ p 4 ](− 65 63 ∑p−1 k=0 ( 2k k )( 4k 2k m k For instance, for any prime p > 7, ) (mod p) and ∑p−1 k=0 ( 2k k )2( ) 4k 2k mk (mod p 2 ). ](− 5 3 ), (1.9) p−1 ∑ k=0 ( 2k k )2(...

3 p

by Zhi-hong Sun
"... Let p> 3 be a prime, and let Rp be the set of rational numbers whose denominator is not divisible by p. Let {Pn(x)} be the Legendre polynomials. In this paper we mainly show that for m, n, t ∈ Rp with m ≡ 0 (mod p), ..."
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Let p> 3 be a prime, and let Rp be the set of rational numbers whose denominator is not divisible by p. Let {Pn(x)} be the Legendre polynomials. In this paper we mainly show that for m, n, t ∈ Rp with m ≡ 0 (mod p),

K,

by Zhi-wei Sun , 1102
"... Abstract. Here I give a full list of my conjectures on series for powers of π and other important constants scattered in some of my public preprints. The list contains totally 92 open conjectural series, 87 of which are for 1/π. 1. Various series for some constants ..."
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Abstract. Here I give a full list of my conjectures on series for powers of π and other important constants scattered in some of my public preprints. The list contains totally 92 open conjectural series, 87 of which are for 1/π. 1. Various series for some constants
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