Pairs of equiperimeter and equiareal triangles whose sides are perfect squares

Ajai Choudhry; Arman Shamsi Zargar

doi:10.7169/facm/1985

Pairs of equiperimeter and equiareal triangles whose sides are perfect squares

Functiones et Approximatio Commentarii Mathematici ◽

10.7169/facm/1985 ◽

2021 ◽

Vol -1 (-1) ◽

Author(s):

Ajai Choudhry ◽

Arman Shamsi Zargar

Keyword(s):

Perfect Squares

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84.38 Digital Patterns in Perfect Squares

The Mathematical Gazette ◽

10.2307/3621666 ◽

2000 ◽

Vol 84 (500) ◽

pp. 289

Author(s):

Florian Luca

Keyword(s):

Perfect Squares

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Magic squares of squares over a finite field

Topology, Geometry, and Dynamics - Contemporary Mathematics ◽

10.1090/conm/773/15536 ◽

2021 ◽

pp. 111-122

Author(s):

Stewart Hengeveld ◽

Giancarlo Labruna ◽

Aihua Li

Keyword(s):

Finite Field ◽

Prime Numbers ◽

Similar Question ◽

Magic Squares ◽

Content Type ◽

Magic Square ◽

Twin Primes ◽

Quadratic Residues ◽

Perfect Squares ◽

Open Question

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.

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A Note on Some Perfect Squared Squares

Canadian Journal of Mathematics ◽

10.4153/cjm-1951-036-9 ◽

1951 ◽

Vol 3 ◽

pp. 304-308 ◽

Cited By ~ 10

Author(s):

T. H. Willcocks

Keyword(s):

Finite Number ◽

Perfect Squares

In a recent paper [5], general methods were described for the dissection of a square into a finite number n of unequal non-overlapping squares. In this note, examples of such perfect squares are given in which the sides and elements are relatively small integers; in particular, a dissection of a square into 24 different elements, which is believed to be the squaring of least order known at the present time.

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