Abstract
Euclid’s proof that the prime numbers are “more than any assigned multitude” (Elements, proposition IX, 20) has long been hailed as a model of elegance and simplicity. Yet, surprisingly, it has also been misrepresented in a great many accounts: The article Hardy and Woodgold (2009) gives a detailed list of sources, including many by eminent number theorists, that either erroneously describe the structure of Euclid’s proof or make false historical claims about it. It is wise, therefore, to begin by quoting Euclid’s argument directly, as it is given in Heath’s translation (Heath 1956, vol. II, p. 412).
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Notes
- 1.
If \(p_{1},p_{2},\ldots,p_{k}\) are the first k primes, then since \(p_{1}p_{2}\ldots p_{k} + 1 < 2 \cdot p_{1}p_{2}\ldots p_{k}\), one can show by induction that \(p_{k+1} \leq 2^{(2^{k}) }\).
- 2.
In 2008, essentially the same proof appeared in the American Mathematical Monthly (Cusumano et al. 2008), without reference to Stieltjes.
- 3.
Euclid’s proof is by reductio, but the contrapositive, that if D divides any two of A, B, A + B it must divide the third as well, is easily proven directly.
- 4.
But see Aigner and Ziegler (2000) for the ingenious, relatively short proof of Bertrand’s Postulate given in 1932 by Paul Erdős (his first publication).
- 5.
As noted above in footnote 1, Euclid’s proof can be analyzed to yield such a lower bound as well.
- 6.
Kronecker replaced n and p by n A and p A, respectively, with A > 1, thereby obtaining a convergent series.
- 7.
Of course, in Euclid’s proof one could just as well let \(N = L - 1\) instead of L + 1.
- 8.
E.g., if n ≥ 2, then none of the n − 1 numbers \(n! + 2,n! + 3,\ldots,n! + n\) can be prime.
References
Aigner, M., Ziegler, G.M.: Proofs from the Book, 2nd ed. Springer, Berlin (2000)
Cusumano, A., Dudley, U., Dodge, C: A variation on Euclid’s proof of the infinitude of the primes. Amer. Math. Monthly 115(7), 663 (2008)
Erdős, P.: Über die Reihe Σ1∕p. Math. Zutphen B 7, 1–2 (1938)
Furstenberg, H.: On the infinitude of primes. Amer. Math. Monthly 62(5), 353 (1955)
Hardy, G.H., Wright, G.M.: Introduction to the Theory of Numbers. Clarendon, Oxford (1938)
Hardy, M., Woodgold, C.: Prime simplicity. Math. Intelligencer 31(4), 44–52 (2009)
Heath, T.: The Thirteen Books of Euclid’s Elements (3 vols.). Dover, New York (1956)
Narkiewicz, W.: The Development of Prime Number Theory. Springer, Berlin (2000)
Niven, I.: A proof of the divergence of Σ1∕p. Amer. Math. Monthly 78(3), 272–273 (1971)
Stieltjes, T.J.: Sur la théorie des nombres. Ann. fac. sciences Toulouse IV(1)(3), 1–103 (1890)
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Dawson, J.W. (2015). The Infinitude of the Primes. In: Why Prove it Again?. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17368-9_7
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