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Lagrange's four-square theorem
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==References== *{{Cite book | last1 = Hardy | first1 = G. H. | author1-link = G. H. Hardy | last2 = Wright | first2 = E. M. | author2-link = E. M. Wright | editor1-last = Heath-Brown | editor1-first = D. R. | editor1-link = Roger Heath-Brown | editor2-last = Silverman | editor2-first = J. H. | editor2-link = Joseph H. Silverman | editor3-last = Wiles | editor3-first = Andrew | editor3-link = Andrew Wiles | title = An Introduction to the Theory of Numbers | edition = 6th | publisher = Oxford University Press | year = 2008 | orig-year = 1938 | isbn = 978-0-19-921985-8 }} *{{Cite book | last1 = Ireland | first1 = Kenneth | last2 = Rosen | first2 = Michael | title = A Classical Introduction to Modern Number Theory | publisher = Springer | year = 1990 | edition = 2nd | isbn = 978-1-4419-3094-1 | doi = 10.1007/978-1-4757-2103-4 }} *{{Cite book | last = Landau | first = Edmund | author-link = Edmund Landau | edition = 2nd | translator-last = Goodman | translator-first = Jacob E. | title = Elementary Number Theory | publisher = AMS Chelsea Publishing | year = 1958 | orig-year = 1927 | volume = 125 | url = http://bookstore.ams.org/chel-125/ }} *{{Cite book | last1 = Niven | first1 = Ivan | last2 = Zuckerman | first2 = Herbert S. | title = An introduction to the theory of numbers | publisher = [[John Wiley & Sons|Wiley]] | year = 1960 }} *{{Cite journal | last = Oh | first = Byeong-Kweon | title = Representations of Binary Forms by Quinary Quadratic Forms | journal= Trends in Mathematics | year = 2000 | volume = 3 | number = 1 | pages = 102β107 | url = http://trends.mathnet.or.kr/mathnet/kms_tex/974363.pdf }} *{{Cite journal | last1 = Rabin | first1 = M. O. | author1-link = Michael O. Rabin | last2 = Shallit | first2 = J. O. | author2-link = Jeffrey Shallit | title = Randomized Algorithms in Number Theory | journal = Communications on Pure and Applied Mathematics | volume = 39 | year = 1986 | number = S1 | pages = S239βS256 | doi = 10.1002/cpa.3160390713 }} *{{Cite journal | last1=Ramanujan | first1=S. | authorlink1=Srinivasa Ramanujan | title=On the expression of a number in the form ''ax''<sup>2</sup> + ''by''<sup>2</sup> + ''cz''<sup>2</sup> + ''du''<sup>2</sup> | journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] | year=1916 | volume=19 | pages=11β21 | url=https://archive.org/details/proceedingsofcam1920191721camb/page/n23/mode/2up}} *{{Cite web | last = Sarnak | first = Peter | author-link = Peter Sarnak | title = The Ramanujan Conjecture and some Diophantine Equations | website = [[YouTube]] | url = https://www.youtube.com/watch?v=1GtOMVcfjJ0 | year = 2013 | type = Lecture at Tata Institute of Fundamental Research | series = ICTS Lecture Series | location = Bangalore, India }} *{{Cite book | last = Stillwell | first = John | author-link = John Stillwell | title = Elements of Number Theory | publisher = Springer | year = 2003 | isbn = 978-0-387-95587-2 | zbl = 1112.11002 | doi = 10.1007/978-0-387-21735-2 | series = Undergraduate Texts in Mathematics }} *{{Cite journal | last = Sun | first = Z.-W. | author-link = Zhi-Wei Sun | title = Refining Lagrange's four-square theorem | journal = J. Number Theory | volume = 175 | year = 2017 | pages = 167β190 | doi = 10.1016/j.jnt.2016.11.008 | arxiv = 1604.06723 | s2cid = 119597024 }} *{{Cite book | last = Williams | first = Kenneth S. | title = Number theory in the spirit of Liouville | series = London Mathematical Society Student Texts | volume = 76 | publisher = [[Cambridge University Press]] | year = 2011 | isbn = 978-0-521-17562-3 | zbl = 1227.11002 }} *{{Cite book | last=Spencer | first=Joel | date=1996 | pages=295β297 | publisher=Springer US | language=en | doi=10.1007/978-1-4612-2418-1_22 | isbn=9780387948263 | title=Number Theory: New York Seminar 1991β1995 | chapter=Four Squares with Few Squares }} *{{cite journal |first1=P. |last1=Pollack |first2=E. |last2=TreviΓ±o |title=Finding the four squares in Lagrange's theorem |journal=Integers |volume=18A |year=2018 |pages=A15 |url=http://math.colgate.edu/~integers/sjs15/sjs15.pdf}}
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