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<p>We give elementary proofs of some congruence criteria to compute binomial coefficients modulo a prime number. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas’s Theorem by using those criteria. We give applications of these criteria by describing a method to derive identities and congruences involving sums of binomial coefficients.</p>

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An extension of Lucas’s Theorem

  • 期刊论文

  • 发表日期：

    2025年10月31日

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  • Zubeyir Cinkir,
  • Aysegul Ozturkalan

摘要

We give elementary proofs of some congruence criteria to compute binomial coefficients modulo a prime number. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas’s Theorem by using those criteria. We give applications of these criteria by describing a method to derive identities and congruences involving sums of binomial coefficients.