Let {Fₙ} be the Fibonacci sequence defined by F₀=0, F₁=1, $F_{n+1}=Fₙ+F_{n-1} (n≥1)$. It is well known that $F_{p-(5/p)}≡ 0 (mod p)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p²|F_{p-(5/p)}$ is always impossible; up to now this is still open. In this paper the sum $∑_{k≡ r (mod 10)}{n\choose k}$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient $F_{p-(5/p)}/p$ and a criterion for the relation $p|F_{(p-1)/4}$ (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative answer to Wall's question implies the first case of FLT (Fermat's last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonacci primes or Lucas primes.
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Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $∑_{k=0}^{p^{a}-1} (hp^{a}-1 \atop k) (2k \atop k)/m^{k} (mod p²)$, where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and $p^{a} > 3$, then $∑_{k=0}^{p^{a}-1} (hp^{a}-1 \atop k)(2k \atop k)(-h/2)^{k} ≡ ((1-2h)/(p^{a}))(1 + h((4-2/h)^{p-1} - 1)) (mod p²)$, where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If $p^{a} > 3$ then $∑_{k=0}^{p^{a}-1} (p^{a}-1 \atop k)(2k \atop k)(-1)^{k} ≡ 3^{p-1} (p^{a}/3) (mod p²)$.
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For k = 1,2,... let $H_k$ denote the harmonic number $∑_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have $∑_{k=1}^{p-1} (H_k)/(k2^k) ≡ 7/24 pB_{p-3} (mod p²)$, $∑_{k=1}^{p-1} (H_{k,2})/(k2^k) ≡ - 3/8 B_{p-3} (mod p)$, and $∑_{k=1}^{p-1} (H²_{k,2n})/(k^{2n}) ≡ (\binom{6n+1}{2n-1} + n)/(6n+1) pB_{p-1-6n} (mod p²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}: = ∑_{j=1}^k 1/(j^m)$.
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For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = {0,1,2,...} has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as $p₃(x) + p₅(y) + p_{11}(z)$ and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in contrast, we show that $p₃ + p₅ + p_{11}$ and 3p₃ + p₅ + p₇ are universal over ℤ. Our proofs are essentially elementary and hence suitable for general readers.
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Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p-1} ≡ (p/3) 3^{p-1} (mod p²)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is $∑_{k=0}^{p-1} \binom{p-1}{k}\binom{2k}{k} ((-1)^k - (-3)^{-k}) ≡ (p/3)(3^{p-1} - 1) (mod p³)$. In addition, we get some new combinatorial identities.
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