Prime Curios!: 105

105
(another Prime Pages' Curiosity)

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Paul Erdős conjectured that this is the largest number n such that the positive values of n - 2^k are all prime.

105/pi⁴ = (1 + 1/2⁴)(1 + 1/3⁴)(1 + 1/5⁴)(1 + 1/7⁴)(1 + 1/11⁴) .... [Ramanujan]

The first product of 3 distinct odd primes p. Note that p +/- 2 are primes. [Gallardo]

105 is the first degree for which the cyclotomic polynom factor are not all 1, 0 or -1. [Cristian]

The largest natural number n for which all odd numbers k relatively prime to n, with 1 < k < n, are primes. It is also the largest for which 2n-p is prime for each prime p in [n, 2n-2]. [Capelle]

(There is one curio for this number that has not yet been approved by an editor.)

To link to this page use http://primes.utm.edu/curios/page.php?number_id=28