On Formulae for the Nth Prime Number

• Published in 1964
• Added on 2026-01-12

Let \(p_n\) denote the nth prime number. (\(p_1=2\), \(p_2 = 3\), etc.) Let \([x]\) denote the greatest integer which is not greater than \(x\). From Wilson’s theorem, \(\frac{(x-1)!+1}{x}\) is an integer for \(x = 1\) and for all prime values of \(x\); but is fractional for all composite values of \(x\).

Links

• https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/on-formulae-for-the-nth-prime-number/43E49D11DFEAD3E4CBC12F17C87F5EE1
• https://www.cambridge.org/core/services/aop-cambridge-core/content/view/43E49D11DFEAD3E4CBC12F17C87F5EE1/S0025557200051366a.pdf/div-class-title-on-formulae-for-the-span-class-italic-n-span-th-prime-number-div.pdf

Other information

BibTeX entry

@article{OnFormulaefortheNthPrimeNumber,
	key = {OnFormulaefortheNthPrimeNumber},
	type = {article},
	title = {On Formulae for the Nth Prime Number},
	author = {C. P. Willans},
	abstract = {Let \(p{\_}n\) denote the nth prime number. (\(p{\_}1=2\), \(p{\_}2 = 3\), etc.)

Let \([x]\) denote the greatest integer which is not greater than \(x\). From Wilson’s theorem, \(\frac{\{}(x-1)!+1{\}}{\{}x{\}}\) is an integer for \(x = 1\) and for all prime values of \(x\); but is fractional for all composite values of \(x\).},
	comment = {},
	date_added = {2026-01-12},
	date_published = {1964-01-12},
	urls = {https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/on-formulae-for-the-nth-prime-number/43E49D11DFEAD3E4CBC12F17C87F5EE1,https://www.cambridge.org/core/services/aop-cambridge-core/content/view/43E49D11DFEAD3E4CBC12F17C87F5EE1/S0025557200051366a.pdf/div-class-title-on-formulae-for-the-span-class-italic-n-span-th-prime-number-div.pdf},
	collections = {fun-maths-facts,integerology},
	url = {https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/on-formulae-for-the-nth-prime-number/43E49D11DFEAD3E4CBC12F17C87F5EE1 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/43E49D11DFEAD3E4CBC12F17C87F5EE1/S0025557200051366a.pdf/div-class-title-on-formulae-for-the-span-class-italic-n-span-th-prime-number-div.pdf},
	year = 1964,
	urldate = {2026-01-12},
	identifier = {doi:10.2307/3611701},
	journal = {The Mathematical Gazette},
	publisher = {Cambridge University Press},
	volume = 48,
	issue = 366,
	issn = {2056-6328},
	doi = {10.2307/3611701},
	pages = {413-415}
}