A sum of two squares just below every large number
Subtracting from $X$ the largest square $u^2 \le X$, and then the largest square $v^2 \le X - u^2$, shows that there is a sum of two squares within $O(X^{1/4})$ of $X$. Green and Lindqvist, by a different argument, found within $O(X^{1/4})$ of $X$ a number $u^2 + v^2$ with $u$ and $v$ almost equal.
Problem. Does the interval $[X - \frac{1}{10} X^{1/4}, X]$ always contain a sum of two squares?
Source: An old problem: J. E. Littlewood, Some problems in real and complex analysis (1968), Problem 2; P. Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), Problem 15; H. L. Montgomery, Ten lectures (1994). Listed as Problem 66 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- NT — Number Theory
- Keywords
- gaps, Littlewood's problems, short intervals, sums of two squares
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{erdos1957,
author = {Erd{\H o}s, Paul},
title = {Some unsolved problems},
journal = {Michigan Mathematical Journal},
volume = {4},
number = {3},
pages = {291--300},
year = {1957},
doi = {10.1307/mmj/1028997963}
}
@book{littlewood1968,
author = {Littlewood, J. E.},
title = {Some Problems in Real and Complex Analysis},
series = {Heath Mathematical Monographs},
publisher = {D. C. Heath},
year = {1968}
}
@book{montgomery1994,
author = {Montgomery, Hugh L.},
title = {Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis},
series = {CBMS Regional Conference Series in Mathematics},
volume = {84},
publisher = {American Mathematical Society},
year = {1994},
doi = {10.1090/cbms/084}
}
@article{bambah1947,
author = {Bambah, R. P. and Chowla, S.},
title = {On numbers which can be expressed as a sum of two squares},
journal = {Proceedings of the National Institute of Sciences of India},
volume = {13},
pages = {101--103},
year = {1947}
}
@article{uchiyama1965,
author = {Uchiyama, Sabur{\^o}},
title = {On the distribution of integers representable as a sum of two $h$-th powers},
journal = {Journal of the Faculty of Science, Hokkaido University, Series I},
volume = {18},
number = {3-4},
pages = {124--127},
year = {1965},
doi = {10.14492/hokmj/1530691483}
}
@article{green2019lindqvist,
author = {Green, Ben and Lindqvist, Sofia},
title = {Monochromatic solutions to $x + y = z^2$},
journal = {Canadian Journal of Mathematics},
volume = {71},
number = {3},
pages = {579--605},
year = {2019},
doi = {10.4153/CJM-2017-036-1}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 66; version of January 2026, most recent update December 2025},
year = {2026}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00949, "A sum of two squares just below every large number", clopen.solutions.