# 2357

This number is a prime.

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The fair approximation log 2/3 = log 5/7 between the
successive digits of prime 2357 is a direct consequence of
the observation that 2^{7} is close to
5^{3}. [Beedassy]

The number with each prime digit *d* repeated *d*
times and the whole sandwiched between two blocks of prime
2357, is prime: __2357__22333555557777777__2357__. [Beedassy]

Replacing each prime digit in 2357 by its complement forms
the prime 8753. Note that combining the two primes by
interweaving their digits forms another prime:
8__2__7__3__5__5__3__7__. [Beedassy]

The sum of 2357 and its successive right-truncations (2357 + 235 + 23 + 2 + 0) is prime, and so is the sum of the successive deleted digits (0 + 7 + 5 + 3 + 2). [Beedassy]

2^{p7} +
3^{p5} +
5^{p3} +
7^{p2} is prime, where
*p _{n}* is the

*n*th prime. [Beedassy]

The double Mersenne numbers
*M*_{Mp} =
2^{Mp} - 1, (where
*M*_{p} = 2^{p} - 1) are
primes only for *p* = 2, 3, 5, 7. [Beedassy]

Adding the prime digits (2, 3, 5, 7) either to all primes with a prime number of distinct prime digits (23, 37, 53, 73, 257, 523) or to all nonprimes with a nonprime number of distinct prime digits (2375, 2537, 2573, 2735, 3275, 3572, 3725, 3752, 5327, 5372, 5723, 5732, 7235, 7325, 7352, 7532 ) forms a prime in each case (983 ; 76159) the reversal of whose product is also prime (79246847). [Beedassy]

2357 can be expressed as the repdigit sum of its (prime) digits: 2222 + 3 + 55 + 77. Note that the latter summands concatenate, in some appropriate order, into a prime, in three different ways all starting with "55": 553222277, 552222773, 557722223. [Beedassy]

The alternating product-series 235*7 +/- 23*57 -/+ 2*357 +/- 23*5*7 -/+ 2*35*7 +/- 2*3*57 -/+ 2*3*5*7 yield prime sums (2689, 601) whose concatenations (2689601, 6012689) are also prime. [Beedassy]

The reverse concatenation of the two prime derangements (5273 and 7523) of 2357 is prime: 75235273. Note that sandwiching the latter between two blocks of prime 2357, followed by halving at the middle yields a new pair of primes (23577523 ; 52732357 ) both remaining prime when every digit is replaced by its respective complement (87533587 ; 58378753). [Beedassy]

Replacing from the left the first digits in 2357 by their partial sums forms a succession of four primes,*viz.,* 2357, 557, 107, 17. Note that the reversals of both semiprimes 2357*557 and 107*17 are prime (9482131, 9181). [Beedassy]