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

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.

### 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

TELOS
Full Text:

### References:

 [1] Crandall It. E., Topics in advanced scientific cnmputation (1996) [2] Crandall R., Math. Comp. 66 (217) pp 433– (1997) · Zbl 0854.11002 [3] Ireland K., A claassical introduction to modern number theory,, 2. ed. (1990) · Zbl 0712.11001
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