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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

Software:

TELOS
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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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