Super congruences and Euler numbers

作者：Sun Zhi Wei^*

来源：Science China Mathematics, 2011, 54(12): 2509-2535.

DOI：10.1007/s11425-011-4302-x

• central binomial coefficients
• super congruences
• Euler numbers

Let p > 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that
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where E(0), E(1), E(2), ... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for pi(2), pi(-2) and the constant K := Sigma(infinity)(k=1)(k/3)/k(2) ( with (-) the Jacobi symbol), two of which are
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• 出版日期2011-12
• 单位南京大学