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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.

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 |

### Software:

TELOS### 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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