Congruences for Franel numbers
Sun, Zhi-Wei
Original · EN
The Franel numbers given by fₙ=∑ₖ₌₀ⁿnk³ (n=0,1,2,) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel numbers. We mainly establish for any prime p>3 the following congruences: align*∑ₖ₌₀ᵖ⁻¹(-1)ᵏfₖ&≡(p3)(modp²), ∑ₖ₌₀ᵖ⁻¹(-1)ᵏkfₖ&≡- 23(p3)(modp²), ∑ₖ₌₁ᵖ⁻¹(-1)ᵏkfₖ &≡0(modp²), ∑ₖ₌₁ᵖ⁻¹(-1)ᵏ/k²fₖ&≡0(modp). align*
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