Numbers Differing from Consecutive Squares by Squares

Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

Alon, Angel, Benjamini and Lubetzky [1] recently studied an old problem of Euler on sumsets for which all elements of A+B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B⊂[0,N] with |B|≫(logN)15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B⊂[0,N] with |B|≫(logN)9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank.

Our main aim in this chapter is to determine which integers can be expressed as the sum of a given number of squares, that is, which have the form where each xi e ℤ, for a given k. We shall concentrate mainly on the two most important cases, characterising the sums of two squares, and showing that every non-negative integer is a sum of four squares. We shall adopt two completely different approaches to this problem: the first is mainly algebraic, making use of two number systems, the Gaussian integers and the quaternions; the second approach is geometric, based on the fact that the expression represents the square of the length of the vector (x1,…, xk) in Rk. We shall therefore give two different proofs for several of the main theorems in this chapter. In mathematics, it is often useful to have more than one proof of a result, not because this adds anything to its validity (a single correct proof is enough for this), but rather because the extra proofs may add to our understanding of the result, and may enable us to extend it in different directions.

The D(-1)-quadruple conjecture states that there does not exist a set of four positive integers such that the product of any two distinct elements is one greater than a perfect square. It is proved that if {; ; ; a ; b ; c ; d}; ; ; is such a set, then max{; ; ; a ; b ; c ; d}; ; ; < 10^{; ; ; 10^23}; ; ; , hence a proof of the conjecture is given up to a completely determined, finite (but currently computationally infeasible large) set of possible exceptions to check.

This article shows new properties of positive integers related to the sum of the deviations of the products of dividing the positive number by each of its factors, as well as the positive integer with only three divisors called the perfect square with a square root as a prime number, and the article also shows the building scheme a computer application to find the number of divisors of a number, and to classify the number into prime and composite.

A Diophantine m-tuple with the property D(n), where n is an integer, is defined as a set of m positive integers with the property that the product of its any two distinct elements increased by n is a perfect square. It is known that if n is of the form 4k + 2, then there does not exist a Diophantine quadruple with the property D(n). The author has formerly proved that if n is not of the form 4k + 2 and n is not in {;-15, -12, -7, -4, -3, -1, 3, 5, 7, 8, 12, 13, 15, 20, 21, 24, 28, 32, 48, 60, 84}; ; , then there exist at least two distinct Diophantine quadruples with the property D(n). The main problem of this paper is to consider the set U of all integers n, not of the form 4k + 2, such that there exist at most two distinct Diophantine quadruples with the property D(n). One open question is whether the set U is finite or not. It can be proved that if n in U and |n| > 48, then n can be represented in one of the following forms: 4k + 3, 16k + 12, 8k + 5, 32k + 20. The main results of the this paper are: If n in U {;-9, -1, 3, 7, 11}; ; and n = 3 (mod 4), then the integers |n - 1|/2, |n - 9|/2 and |9n - 1|/2 are primes, and either |n| is prime or n is the product of twin primes. If n in U {;-27, -3, 5, 13, 21, 45}; and n = 5 (mod 8), then the integers |n|, |n - 1|/4, |n - 9|/4 and |9n - 1|/4 are primes.

Let G be a simple graph of order n. A vertex subset is called independent if its elements are pairwise non-adjacent. Two vertices in G are co-neighbour vertices if they share the same neighbours. Clearly, if S is a set of pairwise co-neighbour vertices of a graph G, then S is an independent set of G. Let c=a+b√m and c=a−b√m, where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. In [M. Lepovic, On conjugate adjacency matrices of a graph, Discrete Mathematics, 307, 730-738, 2007], the author defined the matrix Ac(G)=[cij]n to be the conjugate adjacency matrix of G, if cij=c for any two adjacent vertices i and j, cij=c for any two nonadjacent vertices i and j,and cij= 0 if i=j. In [S. Paul, Conjugate Laplacian matrices of a graph, Discrete Mathematics, Algorithms and Applications, 10, 1850082, 2018], the author defined the conjugate Laplacian matrix of graphs and described various properties of its eigenvalues and eigenspaces. In this article, we determine certain properties of the conjugate Laplacian eigenvalues and the eigenvectors of a graph with co-neighbour vertices.

I am most grateful to the Mathematical Gazette for publishing, in the March 1977 issue, my letter about the problem of finding five positive integers such that the sum of any pair is a perfect square. The response far exceeded my expectations and showed that the problem aroused great interest.

A set of m m positive integers { a 1 , a 2 , … , a m } \{a_1, a_2, \ldots , a_m\} is called a Diophantine m m -tuple if a i a j + 1 a_i a_j + 1 is a perfect square for all 1 ≤ i &gt; j ≤ m 1 \le i &gt; j \le m . Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine m m -tuples states that no Diophantine quintuple exists at all. We prove this conjecture.

It is proved that if $k$ and $d$ are positive integers such that the product of any two distinct elements of the set \[ \{F_{2k},\, 5F_{2k},\, 4F_{2k+2},\, d\} \] increased by $4$ is a perfect square, than $d=4L_{2k}F_{4k+2}$. This is a generalization of the results of Kedlaya, Mohanty and Ramasamy for $k=1$.

The simultaneous diophantine equations 5 Y2 − 20 = X2 and 2 Y2 + 1 = Z2

We explore from several perspectives the following question: given a set of integers, X, and a positive integer, N, what is the maximum size, D(X,N), of a subset, A, contained in {1,2,...,N}, before A is forced to contain two distinct elements that differ by an element of X? The set of forbidden differences, X, is called intersective if D(X,N)=o(N), with the most well-studied examples being the set of perfect squares, S, and the set numbers one less than a prime, P-1. In addition to some new results, including exact formulas and estimates for D(X,N) in some non-intersective cases like X=P and X=S+k for some small positive integers, k, we also provide a comprehensive survey of known bounds and extensive computational data. In particular, we utilize an existing algorithm for finding maximum cliques in graphs to determine D(S,N) for all N up to 300, and D(P-1,N) for all N up to 500. None of these exact values appear previously in the literature.

The Cayley Determinant of the Determinant Tensor and the Alon–Tarsi Conjecture

We prove a nontrivial bound of O (| D | 27/56+ε ) for the 3-part of the class number of a quadratic field ℚ(√ D ) by using a variant of the square sieve and the q -analogue of van der Corput's method to count the number of squares of the form 4 x 3 − dz 2 for a square-free positive integer d and bounded x, z .