On Consecutive Quadratic Non-residues: a Conjecture of Issai Schur
نویسنده
چکیده
Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest possible number of consecutive quadratic non-residues modulo p is always less than p. (One can find a brief discussion of this problem in R. K. Guy’s book [5]). Schur also pointed out that the greatest number of consecutive quadratic nonresidues exceeds p for p = 13, since 5, 6, 7, and 8 are all quadratic non-residues (mod p). This paper uses elementary methods to prove the following:
منابع مشابه
On a conjecture of a bound for the exponent of the Schur multiplier of a finite $p$-group
Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $exp(G)$ and let $m=lfloorlog_pk floor$. We show that $exp(M^{(c)}(G))$ divides $exp(G)p^{m(k-1)}$, for all $cgeq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $exp( M(G))$ divides $exp(G)$, for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement...
متن کاملDifference Ramsey Numbers and Issai Numbers
We present a recursive algorithm for finding good lower bounds for the classical Ramsey numbers. Using notions from this algorithm we then give some results for a generalization of the Generalized Schur numbers, which we call Issai numbers.
متن کاملCongruences Related to the Ankeny-artin-chowla Conjecture
Let p be an odd prime with p ⌘ 1 (mod 4) and " = (t + upp)/2 > 1 be the fundamental unit of the real quadratic field K = Q(pp) over the rationals. The Ankeny-Artin-Chowla conjecture asserts that p u, which still remains unsolved. In this paper, we investigate various kinds of congruences equivalent to its negation p | u by making use of Dirichlet’s class number formula, the products of quadrati...
متن کاملOn Silverman's conjecture for a family of elliptic curves
Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...
متن کاملOff-diagonal Generalized Schur Numbers
We determine all values of the 2-colored off-diagonal generalized Schur numbers (also called Issai numbers), an extension of the generalized Schur numbers. These numbers, denoted S(k, l), are the minimal integers such that any red and blue coloring of the integers from 1 to S(k, l) must admit either a solution to ∑k−1 i=1 xi = xk consisting of only red integers, or a solution to ∑l−1 i=1 xi = x...
متن کامل