Between Consecutive Squares

Three 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.

Primes between consecutive squares and the Lindelöf hypothesis

At present it is not known an unconditional proof that between two consecutive squares there is always a prime number. In a previous paper the author proved that, under the assumption of the Lindelöf hypothesis, each of the intervals [n, (n + 1)] ⊂ [1, N ], with at most O(N) exceptions, contains the expected number of primes, for every constant ε > 0. In this paper we improve the result by weak...

Squares from Products of Consecutive Integers

An elliptic surface related to sums of consecutive squares

The theory of Mordell-Weil lattices is applied to a specific example of a rational elliptic surface. This provides a complete description of the sections of this surface, and of the sections which are defined over Q. The surface is related to the diophantine problem of expressing squares as a sum of consecutive squares. Some consequences which our description has to this problem are discussed.

On Sequences of Consecutive Squares on Elliptic Curves

Let C be an elliptic curve defined over Q by the equation y2 = x3+Ax+B where A,B ∈ Q. A sequence of rational points (xi, yi) ∈ C(Q), i = 1, 2, . . . , is said to form a sequence of consecutive squares on C if the sequence of x-coordinates, xi, i = 1, 2, . . ., consists of consecutive squares. We produce an infinite family of elliptic curves C with a 5-term sequence of consecutive squares. Furth...