Prime Curios!: 46

46
(another Prime Pages' Curiosity)

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The 46th prime is 199. Let C = the number of combinations of a group of 199 objects taken 46 at a time. Note that 199!/[46!(199-46)!] has 46 digits and C ± 1 are both primes! [Hartley]

The least even number 2k such that SOP(2, P(k)) is divisible by 2k, i.e., SOP(2, P(k)) is congruent to 0 mod (2k). Note that SOP(2, P(k)) = the sum of primes from 2 to the prime P(k). [Rivera]

There are only 46 known Mersenne primes.

Mersenne wrongly thought 2^67-1 was prime. 46 is the smallest base b for which (b^67-1)/(b-1) is actually prime.

(There are 2 curios for this number that have not yet been approved by an editor.)