# 2357

This number is a prime. 2357 is the smallest prime that contains all of the prime digits. Chinese Emperor Yao was born circa 2357 B.C. (A famous legend holds that he created the game of Go to improve the intelligence of his son.) 22 + 33 + 55 + 77 is prime. [Papazacharias] Letting A = 1, B = 2, ..., Z = 26, then 2357 is the sum of all the values of the U.S. Presidents' last names from Washington to Coolidge. [Trotter] 2357 is also the sum of consecutive primes in at least two ways: (773 + 787 + 797) and (461 + 463 + 467 + 479 + 487). [Trotter] The smallest number, curiously prime, whose square begins with 4 identical (prime) digits: 2357^2 = 5555449. [Post] Inserting nineteen zeros between the four prime digits of 2357 makes another prime of prime length sixtyone. Note that 23 + 57 = 19 + 61 and also 19 + 4 = 23 as 57 + 4 = 61. [De Geest] Twenty-three fifty-seven (23:57) is the largest "prime time" of day on a 24-hour clock in hours and minutes. [Luhn] 21013 + 31013 + 51013 + 71013 is prime. [Luhn, 2002] 219 + 319 + 519 + 719 is prime. [Ngassam] The prime digits united by unities is prime (2131517). [De Geest] (2*3*5*7+2+3+5+7) and (2*3*5*7-2-3-5-7) are both prime. [Poo Sung] 100^n + 10^n - 1 is prime for consecutive primes 2-3-5-7. [Patterson] The product of the primes less than or equal to 2357 is the smallest titanic primorial number. The number of pies used in the shooting of The Great Race (biggest pie throwing scene on film). Oklahoma is the only U.S. state name whose letters in prime positions are all consonants. The fair approximation log 2/3 = log 5/7 between the successive digits of prime 2357 is a direct consequence of the observation that 27 is close to 53. [Beedassy] The number with each prime digit d repeated d times and the whole sandwiched between two blocks of prime 2357, is prime: 2357223335555577777772357. [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: 82735537. [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] 2p7 + 3p5 + 5p3 + 7p2 is prime, where pn is the nth prime. [Beedassy] 2357 = 1234 + 1123. Note the first four prime, natural, and Fibonacci numbers. [Silva] The double Mersenne numbers MMp = 2Mp - 1, (where Mp = 2p - 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] The sum of the cubes of the first four Fibonacci primes. Curiously, the prime contains three of them. [Silva] On the night of 11/13/1991, the Virginia Lottery Pick 4 winning numbers were 2, 3, 5, 7. 2357 is the only integer that is both a concatenate prime and a minimal k-primeval prime. (2+3+5+7)*2*3*5*7 +/- 2357 are both prime. [Bui Quang Tuan] The larger of only two primes less than a googol formed from the concatenation in order of first n consecutive primes, (case n=4). The previous is 23 for n=2. [Loungrides] G. L. Honaker, Jr., asks, "Does the sum of the series 1/2 + 1/3 + 1/5 + 1/7 + 1/23 + 1/37 + ... + 1/2357 + ..., converge?" Update: Chuck Gaydos of Arizona has determined that the limit equals 1.333... . (Is it 4/3 = 1.333... ?) In Prime Number Sudoku, only the primes 2,3,5 and 7 are allowed in greyed cells. 23 is the aliquot sum of 57. (2, 3, 5, 7) is the only set consisting of four consecutive primes, such that the sum of any three of them minus the other is a prime. [Loungrides] The sequence of primes with distinct prime digits (those with 2, 3, 5, 7 only) have sum of digits equal to 2*3*5*7. [Honaker] The polynomial (-X^3 + 9*X^2 - 14*X + 18)/6 produces 2, 3, 5, 7 for x = 1, 2, 3, 4. [Gaydos] (2+3+5-7), (2+3-5+7), (2-3+5+7), (2+3+5+7), (2*3*5-7), (2*3*5+7) and (2+3*5*7) are all prime. [Homewood] The 7 in 2357 is the 666th 7 to appear in a positive integer. [Gaydos] Why not celebrate Prime Number Day on 2/3 from 5 a.m. to 7 p.m.? Can you identify a criminal from a lineup of prime suspects? See slide 14 of the PowerPoint presentation "A Primer on Prime Numbers." The only 4-digit prime whose the square begins with four identical digits, i.e., 2357^2 = 5555449. [Loungrides] In a list of centered hexagonal numbers, the 2nd, 3rd, 5th and 7th are all Mersenne prime exponents. [Homewood] n! * n falls between a pair of prime twins for n = 2, 3, 5, and 7. Is this true for any other n? [Gaydos]

```Prime numbers are 2, 3, 5 and 7,
And so is the whole number 11.
They have no other factors on the shelf,
Except for 1 and the number itself!
```
[SplashLearn]

(There are 15 curios for this number that have not yet been approved by an editor.)