3,861 research outputs found

    Supercongruences involving dual sequences

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    In this paper we study some sophisticated supercongruences involving dual sequences. For n=0,1,2,…n=0,1,2,\ldots define dn(x)=βˆ‘k=0n(nk)(xk)2kd_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k and sn(x)=βˆ‘k=0n(nk)(xk)(x+kk)=βˆ‘k=0n(nk)(βˆ’1)k(xk)(βˆ’1βˆ’xk).s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom nk(-1)^k\binom xk\binom{-1-x}k. For any odd prime pp and pp-adic integer xx, we determine βˆ‘k=0pβˆ’1(Β±1)kdk(x)2\sum_{k=0}^{p-1}(\pm1)^kd_k(x)^2 and βˆ‘k=0pβˆ’1(2k+1)dk(x)2\sum_{k=0}^{p-1}(2k+1)d_k(x)^2 modulo p2p^2; for example, we establish the new pp-adic congruence βˆ‘k=0pβˆ’1(βˆ’1)kdk(x)2≑(βˆ’1)⟨x⟩p(modp2),\sum_{k=0}^{p-1}(-1)^kd_k(x)^2\equiv(-1)^{\langle x\rangle_p}\pmod{p^2}, where ⟨x⟩p\langle x\rangle_p denotes the least nonnegative integer rr with x≑r(modp)x\equiv r\pmod p. For any prime p>3p>3 and pp-adic integer xx, we determine βˆ‘k=0pβˆ’1sk(x)2\sum_{k=0}^{p-1}s_k(x)^2 modulo p2p^2 (or p3p^3 if x∈{0,…,pβˆ’1}x\in\{0,\ldots,p-1\}), and show that βˆ‘k=0pβˆ’1(2k+1)sk(x)2≑0(modp2).\sum_{k=0}^{p-1}(2k+1)s_k(x)^2\equiv0\pmod{p^2}. We also pose several related conjectures.Comment: 35 pages, final published versio

    New observations on primitive roots modulo primes

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    We make many new observations on primitive roots modulo primes. For an odd prime pp and an integer cc, we establish a theorem concerning βˆ‘g(g+cp)\sum_g(\frac{g+c}p), where gg runs over all the primitive roots modulo pp among 1,…,pβˆ’11,\ldots,p-1, and (β‹…p)(\frac{\cdot}p) denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime pp there is a primitive root g<pg<p modulo pp with gβˆ’1g-1 a square, and that for any prime p>3p>3 there is a prime q<pq<p with the Bernoulli number Bqβˆ’1B_{q-1} a primitive root modulo pp. We also make related observations on quadratic nonresidues modulo primes and primitive prime divisors of some combinatorial sequences. For example, based on heuristic arguments we conjecture that for any prime p>3p>3 there exists a Fibonacci number Fk<p/2F_k<p/2 which is a quadratic nonresidue modulo pp; this implies that there is a deterministic polynomial time algorithm to find square roots of quadratic residues modulo a prime p>3p>3.Comment: 23 page

    A result similar to Lagrange's theorem

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    Generalized octagonal numbers are those p8(x)=x(3xβˆ’2)p_8(x)=x(3x-2) with x∈Zx\in\mathbb Z. In this paper we mainly show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for 3535 triples (b,c,d)(b,c,d) with 1≀b≀c≀d1\le b\le c\le d (including (2,3,4)(2,3,4) and (2,4,8)(2,4,8)), we prove that any nonnegative integer can be exprssed as p8(w)+bp8(x)+cp8(y)+dp8(z)p_8(w)+bp_8(x)+cp_8(y)+dp_8(z) with w,x,y,z∈Zw,x,y,z\in\mathbb Z. We also pose several conjectures for further research.Comment: 21 pages, final published versio

    Quadratic residues and related permutations and identities

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    Let pp be an odd prime. In this paper we investigate quadratic residues modulo pp and related permutations, congruences and identities. If a1<…<a(pβˆ’1)/2a_1<\ldots<a_{(p-1)/2} are all the quadratic residues modulo pp among 1,…,pβˆ’11,\ldots,p-1, then the list {12}p,…,{((pβˆ’1)/2)2}p\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p (with {k}p\{k\}_p the least nonnegative residue of kk modulo pp) is a permutation of a1,…,a(pβˆ’1)/2a_1,\ldots,a_{(p-1)/2}, and we show that the sign of this permutation is 11 or (βˆ’1)(h(βˆ’p)+1)/2(-1)^{(h(-p)+1)/2} according as p≑3(mod8)p\equiv3\pmod 8 or p≑7(mod8)p\equiv7\pmod 8, where h(βˆ’p)h(-p) is the class number of the imaginary quadratic field Q(βˆ’p)\mathbb Q(\sqrt{-p}). To achieve this, we evaluate the product ∏1≀j<k≀(pβˆ’1)/2(cot⁑πj2/pβˆ’cot⁑πk2/p)\prod_{1\le j<k\le(p-1)/2}(\cot\pi j^2/p-\cot\pi k^2/p) via Dirichlet's class number formula and Galois theory. We also obtain some new identities for the sine and cosine functions; for example, we determine the exact value of ∏1≀j<k≀pβˆ’1cos⁑πaj2+bjk+ck2p\prod_{1\le j<k\le p-1}\cos\pi\frac{aj^2+bjk+ck^2}p for any a,b,c∈Za,b,c\in\mathbb Z with ac(a+b+c)≑̸0(modp)ac(a+b+c)\not\equiv0\pmod p.Comment: 36 pages, final published versio

    An additive theorem and restricted sumsets

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    Let G be any additive abelian group with cyclic torsion subgroup, and let A, B and C be finite subsets of G with cardinality n>0. We show that there is a numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the elements of B and a numbering {c_i}_{i=1}^n of the elements of C, such that all the sums a_i+b_i+c_i (i=1,...,n) are distinct. Consequently, each subcube of the Latin cube formed by the Cayley addition table of Z/NZ contains a Latin transversal. This additive theorem can be further extended via restricted sumsets in a field

    Some new inequalities for primes

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    For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.Comment: 6 pages. Make Theorem 1.2 genera

    On Motzkin numbers and central trinomial coefficients

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    The Motzkin numbers Mn=βˆ‘k=0n(n2k)(2kk)/(k+1)M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1) (n=0,1,2,…)(n=0,1,2,\ldots) and the central trinomial coefficients TnT_n (n=0,1,2,…)n=0,1,2,\ldots) given by the constant term of (1+x+xβˆ’1)n(1+x+x^{-1})^n have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with nn any positive integer: 2nβˆ‘k=1n(2k+1)Mk2∈Z,\frac2n\sum_{k=1}^n(2k+1)M_k^2\in\mathbb Z, n2(n2βˆ’1)6β€‰βˆ£β€‰βˆ‘k=0nβˆ’1k(k+1)(8k+9)TkTk+1,\frac{n^2(n^2-1)}6\,\bigg|\,\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1}, and also βˆ‘k=0nβˆ’1(k+1)(k+2)(2k+3)Mk23nβˆ’1βˆ’k=n(n+1)(n+2)MnMnβˆ’1.\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.Comment: 21 page

    On sums of primes and triangular numbers

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    We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2^ap-r)/m+T_x with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with related analysis and numerical data. One of our conjectures states that each natural number not equal to 216 can be written in the form p+T_x with x an integer and p a prime or zero; another conjecture asserts that any odd integer n>3 can be written in the form p+x(x+1) with p a prime and x a positive integer

    Products and sums divisible by central binomial coefficients

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    In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that 2(2n+1)binom(2n,n)| binom(6n,3n)binom(3n,n) for every n=1,2,3,... Also, for any nonnegative integers kk and nn we have (2kk)∣(4n+2k+22n+k+1)(2n+k+12k)(2nβˆ’k+1n)\binom {2k}k | \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n and (2kk)∣(2n+1)(2nn)Cn+k(n+k+12k),\binom{2k}k | (2n+1)\binom{2n}nC_{n+k}\binom{n+k+1}{2k}, where CmC_m denotes the Catalan number (2mm)/(m+1)=(2mm)βˆ’(2mm+1)\binom{2m}m/(m+1)=\binom{2m}m-\binom{2m}{m+1}. Applying this result we obtain two sums divisible by central binomial coefficients.Comment: 15 pages. Submitted version. Add some conjectures and references

    The least modulus for which consecutive polynomial values are distinct

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    Let dβ‰₯4d\ge4 and c∈(βˆ’d,d)c\in(-d,d) be relatively prime integers. We show that for any sufficiently large integer nn (in particular n>24310n>24310 suffices for 4≀d≀364\le d\le 36), the smallest prime p≑c(modd)p\equiv c\pmod d with pβ‰₯(2dnβˆ’c)/(dβˆ’1)p\ge(2dn-c)/(d-1) is the least positive integer mm with 2r(d)k(dkβˆ’c)Β (k=1,…,n)2r(d)k(dk-c)\ (k=1,\ldots,n) pairwise distinct modulo mm, where r(d)r(d) is the radical of dd. We also conjecture that for any integer n>4n>4 the least positive integer mm such that $|\{k(k-1)/2\ \mbox{mod}\ m:\ k=1,\ldots,n\}|= |\{k(k-1)/2\ \mbox{mod}\ m+2:\ k=1,\ldots,n\}|=nistheleastprime is the least prime p\ge 2n-1with with p+2$ also prime.Comment: Final published versio
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