We make more accessible a neglected continued fraction algorithm of Lagrange for solving the equation at2 + btu+ cu2 = n in relatively prime integers t, u, where a > 0, gcd(a, n) = 1, and D = b2 − 4ac < 0. The cases D = −4 and D = −3 present a consecutive convergents phenomenon which aids the search for solutions.

Abstract Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in {Z}$ primitive root modulo p . $0&lt;a_1&lt;\cdots &lt;a_n&lt;p$ all the cubic residues in interval $(0,p)$ Then clearly sequence $a_1 \bmod p,\, a_2 p,\ldots , a_n p$ is permutation of $g^3 p,\,g^6 g^{3n} We determine sign this permutation.

Suppose that A ⊂ {1, . . . , N} is such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(−c 4 √ logN)) for some absolute c > 0.

Let q = p with n = 2m and p be an odd prime. Let 0 ≤ k ≤ n − 1 and k 6= m. In this paper we determine the value distribution of following exponential(character) sums

‎‎Let $R$ be a multiplicative hyperring‎. In this paper‎, ‎we introduce and study the concept of n-absorbing hyperideal which is a generalization‎ ‎of prime hyperideal‎. ‎A proper hyperideal $I$ of $R$ is called an $n$-absorbing hyperideal of ‎$‎R‎$‎ if whenever $alpha_1o...oalpha_{n+1} subseteq I$ for $alpha_1,...,alpha_{n+1} in R$‎, ‎then there are $n$ of the $alpha_i^,$s whose product ...

In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let p be a prime and let a be any positive integer. We determine P p a −1 k=0`2k k+d´mod p 2 for a −1 k=0`2k k+δ´mod p 3 for δ = 0, 1. We also show that 1 C n p a −1 X k=0 C p a n+k ≡ 1 − 3(n + 1) " p a − 1 3 « (mod p 2) for every n = 0, 1, 2,. .. , where C m is the Catalan number...

We prove that if xm + axn permutes the prime field Fp, where m > n > 0 and a ∈ Fp, then gcd(m − n, p − 1) > √ p − 1. Conversely, we prove that if q ≥ 4 and m > n > 0 are fixed and satisfy gcd(m − n, q − 1) > 2q(log log q)/ log q, then there exist permutation binomials over Fq of the form xm + axn if and only if gcd(m,n, q − 1) = 1.

In [17] Lee and Shiue showed that if R is a non-commutative prime ring, I a nonzero left ideal of R and d is a derivation of R such that [d(x)x, x]k = 0 for all x ∈ I, where k,m, n, r are fixed positive integers, then d = 0 unless R ∼= M2(GF (2)). Later in [1] Argaç and Demir proved the following result: Let R be a non-commutative prime ring, I a nonzero left ideal of R and k,m, n, r fixed posi...

Let G = EA(g) of order g be the abelian group ZPl X ZPl X . .. X ZPl X ... X ZPn X ZPn X ... X ZPII n whereZpi occurs ri times with IT pp the prime decomposition of g. i = 1 It is shown that the necessary conditions A==O(modg) v?:: 3n v == 0 (mod n) A(V n) == 0 (mod 2) v v n 0 (mod 24) if g is even, A ( _ ) = (0 (mod 6) if g is odd, are sufficient for the existence of a PGBRD(v, 3, A, n; EA(g)).

Let n be a positive integer and let S be a sequence of n integers in the interval [0, n − 1]. If there is an r such that any nonempty subsequence with sum ≡ 0 (mod n) has length = r, then S has at most two distinct values. This proves a conjecture of R. L. Graham. A previous result of P. Erdős and E. Szemerédi shows the validity of this conjecture if n is a large prime number.