Picture of a square

Square of Squares Tables

Andrew Bremner's article on squares of squares included the 3x3 square:

Bremmer's square

The numbers in the right diagonal as the tuple (972,1132,1272) appears to have come out of the blue. But I will show that this sequence is part of a larger set of tuples having the same property, i.e. the first number in the tuple when added to a difference (Δ) gives the second square in the tuple and when this same (Δ) is added to the second square produces a third square. All these tuple sequences can be used as entries into the right diagonal of a magic square.

It was shown previously that these numbers are a part of a sequence of squares and this page is a continuation of that effort.

I will show from scratch, (i.e. from first principles) that these tuples (a2,b2,c2) whose sum a2 + b2 + c23b2 = 0 are generated from another set of tuples that obeys the equation a2 + b2 + c23b20.

Generation of Sequences for the Production of Tables

The diagonal of a magic square consisting of square entries has the sum a2 + b2 + c23b2 = 0. If we treat the first entry a2 as ±a2 then the equation may be written as ±a2 + c2 = 2b2 which on rearrangement gives c2 = 2b2 ± a2 where a can be set initially to any number greater than 0. Let us set a = 1 and proceed as follows.

Table Tr consists the first seven tuples found using the formula c2 = 2b2 − 7. The second table Ti which will be used in the generation of complex squares, i.e., those with real and imaginary coefficients, consists of the first seven tuples found using the formula c2 = 2b2 + 7.

The desired c2 is calculated by searching all b numbers between 1 and 100,000. However, it was found that the ratio of bn+2/bn or cn+2/cn converges on (1 + √2)2 as the b's or c's get larger. This means that moving down every other row on the table each integer value takes on the previous bn or cn multiplied by (1 + √2)2, i.e., 5.8284271247...

In addition the entries in the columns may be ±1 giving eight combinations of (+ + +), (+ + −), (+ − +), (− + +), (+ − −), (− + −), (− − +) and (− − −) to generate eight Tuple tables. Each row in a tuple table will be used to generate secondary tables that will produce the requisite squares (a2,b2,c2) whose sum a2 + b2 + c23b2 = 0. For example taking the tuple (√7, 2, 1) will generate a sequence of tuples (a,b,c) which will obey the latter equation as is shown in square sequence table Part IIA.

If we look at the entries in the columns as a sequence of numbers, then the sequence of positive numbers from Table Tr columns 2 and column 3, the first sequence, bn, has no Sloane number while cn has Sloane number A077446 number. For table Ti the first sequence, bn, also has no Sloane number while cn has the Sloane number A077443.

Table Tr
±√7 ±2±1
±√7 ±4±5
±√7 ±8±11
±√7 ±22±31
±√7 ±46±65
±√7 ±128±181
±√7 ±268±379
±√7 ±746±1055
±√7 ±1562±2209
±√7 ±4348±6149
±√7 ±9104±12875
±√7 ±25342±35839
±√7 ±53062±75041
Table Ti
±√7 i±1±3
±√7 i±3±5
±√7 i±9±13
±√7 i±19±27
±√7 i±53±75
±√7 i±111±157
±√7 i±309±437
±√7 i±647±915
±√7 i±1801±2547
±√7 i±3771±5333
±√7 i±10497±14845
±√7 i±21979±31803
±√7 i±61181±86523

This concludes the introduction to the sequence table page.
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Copyright © 2011 by Eddie N Gutierrez. E-Mail: