Nested squares and evaluations of integer products. (English) Zbl 0967.11003

Dilcher, Karl

Exp. Math. 9, No. 3, 369-372 (2000).

R. E. Crandall [Projects in scientific computation. TELOS, The Electronic Library of Science, Springer, New York (1996; Zbl 0791.65001)] discovered the algebraic identity \[ ((x^2-85)^2- 4176)^2- 2880^2= (x^2-1^2) (x^2-7^2) (x^2-11^2) (x^2-13^2), \] which allows the product of eight integers (on the right) to be evaluated by a succession of three squarings and three subtractions (on the left). The author observes that Crandall’s formula depends on the fact that \(2\cdot 85= 7^2+11^2= 1^2+ 13^2\), and shows that there are infinitely many formulas of Crandall’s type with 3 nested squares (you get one whenever an even number can be written as a sum of two squares in at least two ways). He also shows that there are no such identities with more than 3 nested squares.

Reviewer: Tom M.Apostol (Pasadena)

MSC:

11-04	Software, source code, etc. for problems pertaining to number theory
11Y05	Factorization
11C08	Polynomials in number theory

Keywords:

algebraic identities; nested squares

Citations:

Zbl 0953.65001; Zbl 0791.65001

Software:

TELOS

References:

[1]	Crandall It. E., Topics in advanced scientific cnmputation (1996) · doi:10.1007/978-1-4612-2334-4
[2]	Crandall R., Math. Comp. 66 (217) pp 433– (1997) · Zbl 0854.11002 · doi:10.1090/S0025-5718-97-00791-6
[3]	Ireland K., A claassical introduction to modern number theory,, 2. ed. (1990) · Zbl 0712.11001