Pythagorean triple (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Pythagorean triple" in English language version.

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Yekutieli, Amnon (2023), "Pythagorean triples, complex numbers, abelian groups and prime numbers", The American Mathematical Monthly, 130 (4): 321–334, arXiv:2101.12166, doi:10.1080/00029890.2023.2176114, MR 4567419
(Alperin 2005) Alperin, Roger C. (2005), "The modular tree of Pythagoras" (PDF), American Mathematical Monthly, 112 (9): 807–816, CiteSeerX 10.1.1.112.3085, doi:10.2307/30037602, JSTOR 30037602, MR 2179860
Nahin, Paul J. (1998), An Imaginary Tale: The Story of ${\sqrt {-1}}$ , Princeton, New Jersey: Princeton University Press, pp. 25–26, ISBN 0-691-02795-1, MR 1645703
Elkies, Noam (1988), "On A⁴ + B⁴ + C⁴ = D⁴", Mathematics of Computation, 51 (184): 825–835, doi:10.2307/2008781, JSTOR 2008781, MR 0930224

ams.org

Elkies, Noam (1988), "On A⁴ + B⁴ + C⁴ = D⁴", Mathematics of Computation, 51 (184): 825–835, doi:10.2307/2008781, JSTOR 2008781, MR 0930224

archive.org

Beauregard, Raymond A.; Suryanarayan, E. R. (2000), "Parametric representation of primitive Pythagorean triples", in Nelsen, Roger B. (ed.), Proofs Without Words: More Exercises in Visual Thinking, vol. II, Mathematical Association of America, p. 120, ISBN 978-0-88385-721-2, OCLC 807785075
Posamentier, Alfred S. (2010), The Pythagorean Theorem: The Story of Its Power and Beauty, Prometheus Books, p. 156, ISBN 9781616141813.
Carmichael, Robert D. (1915), Diophantine Analysis, John Wiley & Sons

arxiv.org

Bernhart, Frank R.; Price, H. Lee (2005), Heron's formula, Descartes circles, and Pythagorean triangles, arXiv:math/0701624
Yekutieli, Amnon (2023), "Pythagorean triples, complex numbers, abelian groups and prime numbers", The American Mathematical Monthly, 130 (4): 321–334, arXiv:2101.12166, doi:10.1080/00029890.2023.2176114, MR 4567419

books.google.com

Sierpiński, Wacław (2003), Pythagorean Triangles, Dover, pp. iv–vii, ISBN 978-0-486-43278-6
Houston, David (1993), "Pythagorean triples via double-angle formulas", in Nelsen, Roger B. (ed.), Proofs Without Words: Exercises in Visual Thinking, Mathematical Association of America, p. 141, ISBN 978-0-88385-700-7, OCLC 29664480
For the nonexistence of solutions where $a$ and $b$ are both square, originally proved by Fermat, see Koshy, Thomas (2002), Elementary Number Theory with Applications, Academic Press, p. 545, ISBN 9780124211711. For the other case, in which $c$ is one of the squares, see Stillwell, John (1998), Numbers and Geometry, Undergraduate Texts in Mathematics, Springer, p. 133, ISBN 9780387982892.
Proceedings of the Southeastern Conference on Combinatorics, Graph Theory, and Computing, Volume 20, Utilitas Mathematica Pub, 1990, p. 141, ISBN 9780919628700
Sally, Judith D. (2007), Roots to Research: A Vertical Development of Mathematical Problems, American Mathematical Society, pp. 74–75, ISBN 9780821872673.
This follows immediately from the fact that ab is divisible by twelve, together with the definition of congruent numbers as the areas of rational-sided right triangles. See e.g. Koblitz, Neal (1993), Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics, vol. 97, Springer, p. 3, ISBN 9780387979663.
Pickover, Clifford A. (2009), "Pythagorean Theorem and Triangles", The Math Book, Sterling, p. 40, ISBN 978-1402757969
Stillwell, John (2002), "6.6 Pythagorean Triples", Elements of Number Theory, Springer, pp. 110–2, ISBN 978-0-387-95587-2

clarku.edu

mathcs.clarku.edu

Joyce, D. E. (June 1997), "Book X, Proposition XXIX", Euclid's Elements, Clark University

discovermagazine.com

Kim, Scott (May 2002), "Bogglers", Discover: 82, The equation w⁴ + x⁴ + y⁴ = z⁴ is harder. In 1988, after 200 years of mathematicians' attempts to prove it impossible, Noam Elkies of Harvard found the counterexample, 2,682,440⁴ + 15,365,639⁴ + 18,796,760⁴ = 20,615,673⁴.

doi.org

Robson, Eleanor (2002), "Words and Pictures: New Light on Plimpton 322" (PDF), The American Mathematical Monthly, 109 (2): 105–120, doi:10.1080/00029890.2002.11919845, S2CID 33907668
Mitchell, Douglas W. (July 2001), "An Alternative Characterisation of All Primitive Pythagorean Triples", The Mathematical Gazette, 85 (503): 273–5, doi:10.2307/3622017, JSTOR 3622017, S2CID 126059099
MacHale, Des; van den Bosch, Christian (March 2012), "Generalising a result about Pythagorean triples", Mathematical Gazette, 96: 91–96, doi:10.1017/S0025557200004010, S2CID 124096076
Rosenberg, Steven; Spillane, Michael; Wulf, Daniel B. (May 2008), "Heron triangles and moduli spaces", Mathematics Teacher, 101: 656–663, doi:10.5951/MT.101.9.0656
Yekutieli, Amnon (2023), "Pythagorean triples, complex numbers, abelian groups and prime numbers", The American Mathematical Monthly, 130 (4): 321–334, arXiv:2101.12166, doi:10.1080/00029890.2023.2176114, MR 4567419
Voles, Roger (July 1999), "83.27 Integer solutions of $a^{-2}+b^{-2}=d^{-2}$ ", The Mathematical Gazette, 83 (497): 269–271, doi:10.2307/3619056, JSTOR 3619056, S2CID 123267065
Richinick, Jennifer (July 2008), "92.48 The upside-down Pythagorean theorem", The Mathematical Gazette, 92 (524): 313–316, doi:10.1017/s0025557200183275, JSTOR 27821792, S2CID 125989951
(Alperin 2005) Alperin, Roger C. (2005), "The modular tree of Pythagoras" (PDF), American Mathematical Monthly, 112 (9): 807–816, CiteSeerX 10.1.1.112.3085, doi:10.2307/30037602, JSTOR 30037602, MR 2179860
1988 Preprint Archived 2011-08-09 at the Wayback Machine See Figure 2 on page 3., later published as Fässler, Albert (June–July 1991), "Multiple Pythagorean number triples", American Mathematical Monthly, 98 (6): 505–517, doi:10.2307/2324870, JSTOR 2324870
Benito, Manuel; Varona, Juan L. (June 2002), "Pythagorean triangles with legs less than $n$ ", Journal of Computational and Applied Mathematics, 143 (1): 117–126, Bibcode:2002JCoAM.143..117B, doi:10.1016/S0377-0427(01)00496-4 as PDF
Hirschhorn, Michael (November 2011), "When is the sum of consecutive squares a square?", The Mathematical Gazette, 95: 511–2, doi:10.1017/S0025557200003636, ISSN 0025-5572, OCLC 819659848, S2CID 118776198
Goehl, John F. Jr. (May 2005), "Reader reflections", Mathematics Teacher, 98 (9): 580, doi:10.5951/MT.98.9.0580
Elkies, Noam (1988), "On A⁴ + B⁴ + C⁴ = D⁴", Mathematics of Computation, 51 (184): 825–835, doi:10.2307/2008781, JSTOR 2008781, MR 0930224

fau.edu

math.fau.edu

Yiu, Paul (2008), Heron triangles which cannot be decomposed into two integer right triangles (PDF), 41st Meeting of Florida Section of Mathematical Association of America, p. 17
Yiu, Paul (2003), "Recreational Mathematics" (PDF), Course Notes, Dept. of Mathematical Sciences, Florida Atlantic University, Ch. 2, p. 110

harvard.edu

ui.adsabs.harvard.edu

Benito, Manuel; Varona, Juan L. (June 2002), "Pythagorean triangles with legs less than $n$ ", Journal of Computational and Applied Mathematics, 143 (1): 117–126, Bibcode:2002JCoAM.143..117B, doi:10.1016/S0377-0427(01)00496-4 as PDF

jstor.org

Mitchell, Douglas W. (July 2001), "An Alternative Characterisation of All Primitive Pythagorean Triples", The Mathematical Gazette, 85 (503): 273–5, doi:10.2307/3622017, JSTOR 3622017, S2CID 126059099
Voles, Roger (July 1999), "83.27 Integer solutions of $a^{-2}+b^{-2}=d^{-2}$ ", The Mathematical Gazette, 83 (497): 269–271, doi:10.2307/3619056, JSTOR 3619056, S2CID 123267065
Richinick, Jennifer (July 2008), "92.48 The upside-down Pythagorean theorem", The Mathematical Gazette, 92 (524): 313–316, doi:10.1017/s0025557200183275, JSTOR 27821792, S2CID 125989951
(Alperin 2005) Alperin, Roger C. (2005), "The modular tree of Pythagoras" (PDF), American Mathematical Monthly, 112 (9): 807–816, CiteSeerX 10.1.1.112.3085, doi:10.2307/30037602, JSTOR 30037602, MR 2179860
1988 Preprint Archived 2011-08-09 at the Wayback Machine See Figure 2 on page 3., later published as Fässler, Albert (June–July 1991), "Multiple Pythagorean number triples", American Mathematical Monthly, 98 (6): 505–517, doi:10.2307/2324870, JSTOR 2324870
Pagni, David (September 2001), "Fibonacci Meets Pythagoras", Mathematics in School, 30 (4): 39–40, JSTOR 30215477
Elkies, Noam (1988), "On A⁴ + B⁴ + C⁴ = D⁴", Mathematics of Computation, 51 (184): 825–835, doi:10.2307/2008781, JSTOR 2008781, MR 0930224

loc.gov

lccn.loc.gov

Long (1972, p. 48) Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950

maa.org

Robson, Eleanor (2002), "Words and Pictures: New Light on Plimpton 322" (PDF), The American Mathematical Monthly, 109 (2): 105–120, doi:10.1080/00029890.2002.11919845, S2CID 33907668

nctm.org

Rosenberg, Steven; Spillane, Michael; Wulf, Daniel B. (May 2008), "Heron triangles and moduli spaces", Mathematics Teacher, 101: 656–663, doi:10.5951/MT.101.9.0656
Goehl, John F. Jr. (May 2005), "Reader reflections", Mathematics Teacher, 98 (9): 580, doi:10.5951/MT.98.9.0580

niu.edu

math.niu.edu

Sum of consecutive cubes equal a cube, archived from the original on 2008-05-15

oeis.org

Sloane, N. J. A. (ed.), "Sequence A000129 (Pell numbers)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
Sloane, N. J. A. (ed.), "Sequence A237518 (Least primes that together with prime(n) forms a Heronian triangle)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
Sloane, N. J. A. (ed.), "Sequence A001652", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation; Sloane, N. J. A. (ed.), "Sequence A001653", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
Sloane, N. J. A. (ed.), "Sequence A303734", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
Sloane, N. J. A. (ed.), "Sequence A351061 (Smallest positive integer whose square can be written as the sum of n positive perfect squares)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation

psu.edu

citeseerx.ist.psu.edu

(Alperin 2005) Alperin, Roger C. (2005), "The modular tree of Pythagoras" (PDF), American Mathematical Monthly, 112 (9): 807–816, CiteSeerX 10.1.1.112.3085, doi:10.2307/30037602, JSTOR 30037602, MR 2179860

semanticscholar.org

api.semanticscholar.org

Robson, Eleanor (2002), "Words and Pictures: New Light on Plimpton 322" (PDF), The American Mathematical Monthly, 109 (2): 105–120, doi:10.1080/00029890.2002.11919845, S2CID 33907668
Mitchell, Douglas W. (July 2001), "An Alternative Characterisation of All Primitive Pythagorean Triples", The Mathematical Gazette, 85 (503): 273–5, doi:10.2307/3622017, JSTOR 3622017, S2CID 126059099
MacHale, Des; van den Bosch, Christian (March 2012), "Generalising a result about Pythagorean triples", Mathematical Gazette, 96: 91–96, doi:10.1017/S0025557200004010, S2CID 124096076
Voles, Roger (July 1999), "83.27 Integer solutions of $a^{-2}+b^{-2}=d^{-2}$ ", The Mathematical Gazette, 83 (497): 269–271, doi:10.2307/3619056, JSTOR 3619056, S2CID 123267065
Richinick, Jennifer (July 2008), "92.48 The upside-down Pythagorean theorem", The Mathematical Gazette, 92 (524): 313–316, doi:10.1017/s0025557200183275, JSTOR 27821792, S2CID 125989951
Hirschhorn, Michael (November 2011), "When is the sum of consecutive squares a square?", The Mathematical Gazette, 95: 511–2, doi:10.1017/S0025557200003636, ISSN 0025-5572, OCLC 819659848, S2CID 118776198

sjsu.edu

math.sjsu.edu

(Alperin 2005) Alperin, Roger C. (2005), "The modular tree of Pythagoras" (PDF), American Mathematical Monthly, 112 (9): 807–816, CiteSeerX 10.1.1.112.3085, doi:10.2307/30037602, JSTOR 30037602, MR 2179860

tandfonline.com

Kak, S. and Prabhu, M. Cryptographic applications of primitive Pythagorean triples. Cryptologia, 38:215–222, 2014. [1]

umn.edu

conservancy.umn.edu

1988 Preprint Archived 2011-08-09 at the Wayback Machine See Figure 2 on page 3., later published as Fässler, Albert (June–July 1991), "Multiple Pythagorean number triples", American Mathematical Monthly, 98 (6): 505–517, doi:10.2307/2324870, JSTOR 2324870

purl.umn.edu

1988 Preprint Archived 2011-08-09 at the Wayback Machine See Figure 2 on page 3., later published as Fässler, Albert (June–July 1991), "Multiple Pythagorean number triples", American Mathematical Monthly, 98 (6): 505–517, doi:10.2307/2324870, JSTOR 2324870

unirioja.es

Benito, Manuel; Varona, Juan L. (June 2002), "Pythagorean triangles with legs less than $n$ ", Journal of Computational and Applied Mathematics, 143 (1): 117–126, Bibcode:2002JCoAM.143..117B, doi:10.1016/S0377-0427(01)00496-4 as PDF

web.archive.org

1988 Preprint Archived 2011-08-09 at the Wayback Machine See Figure 2 on page 3., later published as Fässler, Albert (June–July 1991), "Multiple Pythagorean number triples", American Mathematical Monthly, 98 (6): 505–517, doi:10.2307/2324870, JSTOR 2324870
Sum of consecutive cubes equal a cube, archived from the original on 2008-05-15

wolfram.com

mathworld.wolfram.com

Weisstein, Eric W., "Rational Triangle", MathWorld

worldcat.org

search.worldcat.org

Beauregard, Raymond A.; Suryanarayan, E. R. (2000), "Parametric representation of primitive Pythagorean triples", in Nelsen, Roger B. (ed.), Proofs Without Words: More Exercises in Visual Thinking, vol. II, Mathematical Association of America, p. 120, ISBN 978-0-88385-721-2, OCLC 807785075
Houston, David (1993), "Pythagorean triples via double-angle formulas", in Nelsen, Roger B. (ed.), Proofs Without Words: Exercises in Visual Thinking, Mathematical Association of America, p. 141, ISBN 978-0-88385-700-7, OCLC 29664480
Hirschhorn, Michael (November 2011), "When is the sum of consecutive squares a square?", The Mathematical Gazette, 95: 511–2, doi:10.1017/S0025557200003636, ISSN 0025-5572, OCLC 819659848, S2CID 118776198