TWO NEW PRIME VARIANT BORDER 7X7 SQUARES

A 7x7 magic square, discovered by A. Johnson, contains only prime numbers and its internal 5x5 and 3x3 squares are also magic at their shared borders. (See square 000 below referenced from Ian Stuart's book¹ and A. Johnson's original paper²). The Johnson prime square is also shown in the previous web page.

This page depicts two other variants but unlike the Johnson square which has 1979 at the center of the square, these squares have 1931, the number prior to 1979 im the Johnson complementary table. Both squares have identical 3x3 and 5x5 squares but differ at the 7x7 border where only the 4 corner numbers are identical. Thus it depends what prime pairs one uses, i.e., there can be more than one square coinsisting of positive prime numbers as shown here.

Construction of a New Prime 7x7 Magic Square

1. Construct the complementary table of prime pairs where each pair of complements, as well as the 25^th number doubled, sums to 3862.
2. First generate the internal 3x3 square is generated using the nine khaki colored numbers whose difference (Δ) between numbers (as shown in the figure above) is 42.
3. Then generate the 5x5 square by filling in the border with the requisite complementary pairs.
4. Finally finish up by filling in the 7x7 border with the requisite complementary pairs, where the S = 13517.

Construction of a Second Variant of a New Prime 7x7 Magic Square

1. Construct the complementary table of prime pairs where each pair of complements, as well as the 25^th number doubled, sums to 3862.
2. Take the 5x5 square constructed above and fill in the 7x7 border with different requisite complementary pairs. The sum again is 13517.

This concludes the new two prime square variants.
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References

1. Ian Stuart: Professors Stuart's Hoard of Mathematical Treasures(2009) Page 192
2. A. W. Johnson, Jr., Journal of Recreational Mathematics 15:2, 1982-83, p. 84