On squares of squares II

The Geometric Disposition of Diophantine Equations

“…The search for a 3 × 3 magic square of squares was popularized by Martin Gardner in 1996 [Gar96]; he attributed the problem to Martin LaBar (1984), though it had been studied by Euler in 1770 and Lucas in 1876 [Boy05]. Andrew Bremner has used the arithmetic of elliptic curves and K3 surfaces to study two related problems: finding 3×3 squares with distinct square entries such that as many as possible of the eight row, columns, and diagonals are equal [Bre99], and finding magic 3 × 3 squares with distinct entries, with as many entries as possible being squares [Bre01]. Our own investigations [BTVA] into the problem of finding 3 × 3 magic squares of squares are inspired by a similar geometric point of view, although we work with surfaces of general type, as explained below.…”

Elliptic Curves in Recreational Number Theory

“…By Pythagoras, a 2 + b 2 = h 2 and the area ∆ = ab/2. The most well-known such triangle is probably the (3,4,5) triangle introduced at school. It has area 6.…”

“…If we scale the (3,4,5) sides by 2, we get (6, 8, 10) with 6 2 + 8 2 = 10 2 and area 24. Scale by 3 and the area goes up by factor of 9 to 54.…”

“…Scale by 3 and the area goes up by factor of 9 to 54. The next smallest integer right-angled triangle is (5,12,13) with area 30. So restricting to integer sides is, perhaps, too restrictive.…”

On Simultaneous Arithmetic Progressions on Elliptic Curves

“…Apparently these papers had their starting point in the featuring of x-a.p. as by-product of a latin square problem (see more on this in [1,2]). However, highly interesting results were sketched in both papers around the relationship between the existence of arithmetic progressions on a certain elliptic curve and its rank.…”