منابع مشابه
On 4 Squares in Arithmetic Progression
x1 − 2x2 + x3 = 0 x2 − 2x3 + x4 = 0 are given by (x1, x2, x3, x4) = (±1,±1,±1,±1). Now, the above variety is an intersection between 2 quadrics in P. In general – i.e., except for the possibility of the variety being reducible or singular – an intersection between 2 quadrics in P is (isomorphic to) an elliptic curve and there is an algorithm that brings the curve to Weierstraß form by means of ...
متن کاملFive Squares in Arithmetic Progression over Quadratic Fields
We give several criteria to show over which quadratic number fields Q( √ D) there should exists a non-constant arithmetic progressions of five squares. This is done by translating the problem to determining when some genus five curves CD defined over Q have rational points, and then using a Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like method, we prove that the only n...
متن کاملSquares in Arithmetic
I. Let Q(N; q; a) denote the number of squares in the arithmetic progression qn+a; n = 1; 2; ; N; and let Q(N) be the maximum of Q(N; q; a) over all non-trivial arithmetic progressions qn + a. It seems to be remarkably diicult to obtain non-trivial upper bounds for Q(N). There are currently two proofs known of the weak bound Q(N) = o(N) (which is an old conjecture of Erdd os) and both are far f...
متن کاملFinding Almost Squares
We study short intervals which contain an " almost square " , an integer n that can be factored as n = ab with a, b close to √ n. This is related to the problem on distribution of n 2 α (mod 1) and the problem on gaps between sums of two squares.
متن کاملThree Consecutive Almost Squares
Given a positive integer n, we let sfp(n) denote the squarefree part of n. We determine all positive integers n for which max{sfp(n), sfp(n+ 1), sfp(n+ 2)} ≤ 150 by relating the problem to finding integral points on elliptic curves. We also prove that there are infinitely many n for which max{sfp(n), sfp(n + 1), sfp(n + 2)} < n.
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2003
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa110-1-1