# 勒讓德猜想

## 部分結果

R·C·贝克（R. C. Baker）、格林·哈曼英语Glyn Harman平茨·亚诺什匈牙利语Pintz János證明了對於所有大的${\displaystyle x}$ 而言，${\displaystyle [x-x^{21/40},\,x]}$ 該區間內總有一個質數。[7]

## 註解

1. ^ 這是從兩個完全平方數的差會是其平方根的事實推導出的。

## 參考資料

1. ^ Stewart, Ian, Visions of Infinity: The Great Mathematical Problems, Basic Books: 164, 2013, ISBN 9780465022403.
2. ^ Bazzanella, Danilo, Primes between consecutive squares (PDF), Archiv der Mathematik, 2000, 75 (1): 29–34 [2024-01-09], MR 1764888, S2CID 16332859, doi:10.1007/s000130050469, （原始内容存档 (PDF)于2017-08-28）
3. ^ Francis, Richard L., Between consecutive squares, Missouri Journal of Mathematical Sciences (University of Central Missouri, Department of Mathematics and Computer Science), February 2004, 16 (1): 51–57, ; see p. 52, "It appears doubtful that this super-abundance of primes can be clustered in such a way so as to avoid appearing at least once between consecutive squares."
4. ^ Heath-Brown, D. R., The number of primes in a short interval (PDF), Journal für die Reine und Angewandte Mathematik, 1988, 1988 (389): 22–63 [2024-01-09], MR 0953665, S2CID 118979018, doi:10.1515/crll.1988.389.22, （原始内容存档 (PDF)于2019-05-02）
5. ^ Selberg, Atle, On the normal density of primes in small intervals, and the difference between consecutive primes, Archiv for Mathematik og Naturvidenskab, 1943, 47 (6): 87–105, MR 0012624
6. ^ A060199
7. ^ Baker, R. C.; Harman, G.; Pintz, J., The difference between consecutive primes, II (PDF), Proceedings of the London Mathematical Society, 2001, 83 (3): 532–562, S2CID 8964027, doi:10.1112/plms/83.3.532
8. ^ Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to ${\displaystyle 4\cdot 10^{18}}$  (PDF), Mathematics of Computation, 2014, 83 (288): 2033–2060, MR 3194140, .