A295262 Primes for "Landau's trick" to prove Bertrand's postulate for n < 4000.

2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001

Offset: 1

Author

Alonso del Arte, Nov 18 2017

Keywords

• nonn
• fini

Comments

Chapter 2 in the Aigner and Ziegler book is devoted to Bertrand's postulate. The proof given starts by showing Bertrand's postulate is true just for n < 4000.

After 2, each prime is less than twice the previous prime. So, even if these were the only primes up to 4002, Bertrand's postulate would still be true for the specified range.

However, these are different from the Bertrand primes (A006992) after 2503, as that sequence requires the very largest prime smaller than twice the previous one, since twice 2503 is 5006 and 5003 is the largest prime less than that.

Erdős Pál used this sequence, with 4001 instead of 5003, in his 1932 proof of Bertrand's postulate, attributing it to Edmund Landau ("einer Bemerkung des Herrn Landau"), which Aigner and Ziegler refer to as "Landau's trick" in their book.

References

• Martin Aigner and Günter M. Ziegler, Proofs from the Book, Second Edition. Berlin (2001): Springer-Verlag, p. 7.

Links

• Erdos Pál, Beweis eines Satzes von Tschebyshef, Acta Sci. Math (Szeged) 5 (1930-1932), p. 198 (in German).

Crossrefs

Cf. A006992.