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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) The smallest prime power of a prime power of a prime (2^2^2). [Woods] The smallest number which is the sum of two distinct primes in two ways, i.e., 16 = 3 + 13 = 5 + 11. [Honaker, 2000] La Symphonie Primordiale (The Prime Numbers Symphony) is a single movement work for orchestra lasting some 16 minutes. "Take a number n in base b, and append one more digit to get the nth prime." The first base for which this puzzle has no solutions is 16. [Hartley] 16 is the only twodigit number m such that m^2^0+1, m^2^1+1, & m^2^2+1 are primes. [Kazemi] 16 is the smallest square which is difference of two prime squares. [Capelle] The smallest square whose reverse is prime. [Capelle] 2^{22} is the smallest number n such that 3 + n, (3*5) + n, (3*5*17) + n, (3*5*17*257) + n, (3*5*17*257*65537) + n are all primes. Note that 3, 5, 17, 257 and 65537 are the known Fermat primes, with the property (F_{0}*F_{1}*F_{2}*...*F_{n1}) + 2 = F_{n} [Capelle] 16^(1+6)+(1+6) is prime. [Silva] sigma(16^16)+16^16 is prime and the length of this prime has the same property. [Firoozbakht] (16) = 1 * 6. This is the smallest number with this property. [Kumar] 16 is probably the only positive integer N such that if the positive integers are listed in rows of N, then the first row with no prime is immediately followed by another row with no prime. If N=16, then the first row with no prime is 1329, 1330, 1331, 1332, 1333, 1334, 1335, 1336, 1337, 1338, 1339, 1340, 1341, 1342, 1343, 1344. The next row is 1345, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, which contains no prime. [Jacobs] The minimal gap (difference) between consecutive primes may be as low as 16, as n approaches infinity.
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