Magic squares of squares over a finite field

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

High rank elliptic curves induced by rational Diophantine triples

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.

Novel Parametric Solutions for the Ideal and Non-Ideal Prouhet Tarry Escott Problem

The present study aims to develop novel parametric solutions for the Prouhet Tarry Escott problem of second degree with sizes 3, 4 and 5. During this investigation, new parametric representations for integers as the sum of three, four and five perfect squares in two distinct ways are identified. Moreover, a new proof for the non-existence of solutions of ideal Prouhet Tarry Escott problem with degree 3 and size 2 is derived. The present work also derives a three parametric solution of ideal Prouhet Tarry Escott problem of degree three and size two. The present study also aimed to discuss the Fibonacci-like pattern in the solutions and finally obtained an upper bound for this new pattern.

5. More triangles and squares

‘More triangles and squares’ explores Diophantine equations, named after the mathematician Diophantus of Alexandria. These are equations requiring whole number solutions. Which numbers can be written as the sum of two perfect squares? Joseph-Louis Lagrange’s theorem guarantees that every number can be written as the sum of four squares, and Edward Waring correctly suggested that there are similar results for higher powers. In 1637, Fermat conjectured that no three positive integers, a, b, and c, can satisfy the equation an+bn=cn, if n is greater than 2. Known as ‘Fermat’s last theorem’, this conjecture was eventually proved by Andrew Wiles in 1995.

2. Multiplying and dividing

‘Multiplying and dividing’ looks at multiples and divisors, focusing on the least common multiple and greatest common divisor of two numbers. We use Euclid’s algorithm as a method for computing the greatest common divisor of two numbers by using the division rule repeatedly. Perfect squares (integers that are the product of two equal integers) feature throughout number theory. Tests are given for divisibility by certain small numbers. An ancient method called ‘casting out nines’, was developed in India in around the year 1000, based on the argument that a number and its digital sum leave the same remainder when divided by 9. We can still use this method to verify the accuracy (or otherwise) of arithmetical calculations.

ON FUNDAMENTAL PRINCIPLES OF THE OPTIMAL NUMBER AND LOCATION OF LOADING BAYS IN URBAN AREAS

The paper is dealing with the problem of finding the optimal number and location of Loading Bays (LBs) for efficient urban last mile deliveries. To solve the problem a multi-parametric model of the idealized urban area is introduced and applied to various instances of a rectangular urban grid structured zones. Multi-parametric approach is used to assess statistically the most relevant number and location of LBs. Computational and graphical results of the idealized model exhibit geometric patterns showing that the optimal Number of LBs (#LB) naturally tends to perfect squares. Moreover, even in case of generalized instances, at a selected number of LBs their distribution is not random but follows specific laws. The optimality is closely related to the prefixed (maximal) walking distance dmax, from the LB to the customer. Based on various simulations the existence and robustness of a descending convex dependence dmax = (#LB) is proven. The results might serve as a decision-making tool to determine the optimal number and location of LBs for any real-life city centre.