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A080169
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Numbers that are cubes of primes of the form 4k+1 (A002144).
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2
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125, 2197, 4913, 24389, 50653, 68921, 148877, 226981, 389017, 704969, 912673, 1030301, 1295029, 1442897, 2571353, 3307949, 3869893, 5177717, 5929741, 7189057, 7645373, 12008989, 12649337, 13997521, 16974593, 19465109, 21253933
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OFFSET
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1,1
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COMMENTS
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a(n) is the sum of two squares in exactly two ways (Fermat). See the Dickson reference, (B) on p. 277. - Wolfdieter Lang, Jan 15 2015
a(n) is the hypotenuse of three and only three right triangles with integral arms.
In 1640 Fermat generalized the 3,4,5 triangle with the theorem: A prime of the form 4n+1 is the hypotenuse of one and only one right triangle with integral arms. The square of a prime of the form 4n+1 is the hypotenuse of two and only two... The cube of three and only three... .
See the Dickson reference, (A) on p. 227.
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.
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LINKS
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FORMULA
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EXAMPLE
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a(2) = 2197 is the hypotenuse of the three triangles 825, 2035, 2197; 845, 2028, 2197; 1547, 1560, 2197.
a(2) = 9^2 + 46^2 = 39^2 + 26^2, and these are the only decompositions. - Wolfdieter Lang, Jan 15 2015
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MATHEMATICA
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Select[Prime[Range[60]], Mod[#, 4] == 1 &]^3 (* Amiram Eldar, Dec 02 2022 *)
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PROG
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(PARI) fermat(n) = { for(x=1, n, y=4*x+1; if(isprime(y), print1(y^3" ")) ) }
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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Edited: New name, part of old one now as a comment. Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 15 2015
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STATUS
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approved
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