Bambah 1947 numbers which can be expressed as

../

theorem_p103: Bambah and Chowla's theorem that for every eps > 0 and all x > x_0(eps) some sum of two integer squares lies between x and x + 2 sqrt(2+eps) x^{1/4}, which gives their gap bound f(x) = O(x^{1/4}).

---

Bambah, R. P. and Chowla, S., On numbers which can be expressed as a sum of two squares. Proc. Nat. Inst. Sci. India 13 (1947), no. 2, 101-103.

Writing b_1 < b_2 < ... for the integers expressible as a sum of two integer squares, the paper asks for a function f(x) such that the interval from x to x + f(x) always contains such an integer for large x, that is, a bound on the gaps b_(n+1) - b_n. The authors note that the classical lattice-point error term P(x) = O(x^(27/82)) gives only f(x) = O(x^(27/82)), and that the conjectural P(x) = O(x^(1/4+eps)) would give f(x) = O(x^(1/4+eps)). Their result, equation (3), is that f(x) = O(x^(1/4)) holds unconditionally by a short elementary argument, with a more precise version stated as the Theorem at the end of section 2 (p. 103). The proof fixes t = floor(sqrt(x)) and compares the two solutions x_1, x_2 of x_1^2 + t^2 = x and x_2^2 + t^2 = x + 2*sqrt(2+eps)*x^(1/4), showing x_2 - x_1 > 1 so that an integer x_3 lies between them and x_3^2 + t^2 is a sum of two squares in the interval. They credit T. Vijayaraghavan with an earlier, less simple proof of the same bound and note that a conjecture on primes congruent to a mod b in the intervals [x, x + x^eps] would, with a = 1 and b = 4, give f(x) = x^eps, which they say shows that (3) is still very far from the probable truth; they ask whether (3) can be improved by elementary arguments (p. 102).

Source: https://insa.nic.in/writereaddata/UpLoadedFiles/PINSA/Vol13_1947_2_Art05.pdf. No notice is printed in the scan, which has no text layer (pages 1 and 3 rendered), and the archive file it was taken from (https://insa.nic.in/writereaddata/UpLoadedFiles/PINSA/Vol13_1947_2_Art05.pdf) carries none; the publisher's site shows only its site-wide footer "Copyright © 2026 Indian National Science Academy. All Rights Reserved" (https://www.insaindia.res.in/), not an article-level line, every other right reserved.

Read status: claims checked for the setting, equation (3) and the Theorem, read clause by clause on the page images of the print, with the proof in section 2 followed; the lattice-point bounds (1) and (2) and the prime conjecture are cited by the paper, not proved. Nothing here is independently reviewed. Result page: theorem_p103.

Bears on. #222: the Theorem (p. 103) gives, for every $\epsilon>0$, consecutive sums of two squares $n_k<n_{k+1}$ with $n_{k+1}-n_k<2\sqrt{2+\epsilon}\,n_k^{1/4}$ once $n_k>x_0(\epsilon)$, an upper bound $O(n_k^{1/4})$ for the differences the problem asks about; the paper gives no lower bound.

Results.

• Theorem (p. 103) and equation (3) (p. 101): for every $\epsilon>0$ and all $x>x_0(\epsilon)$ some sum of two squares lies between $x$ and $x+2\sqrt{2+\epsilon}\,x^{1/4}$; hence $f(x)=O(x^{1/4})$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.