# 16

This number is a composite.

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]

*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 two-digit 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_{n-1})
+ 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]

π(16) = 1 * 6. This is the smallest number with this property. [Kumar]

The smallest square whose sum of digits is a prime number. [Homewood]

Only 2-digit number n which gives prime number for expression 2^n+1 . [Bopardikar]

The first string of 16 numbers with prime factorizations
such that their numbers of divisors are equal most likely
includes a number congruent to 2^{14} modulo
2^{15}. Why? [Merickel]

Add the 16-th prime and the 16-th composite number together to get a prime whose sum of digits is 16. Can you find the next case like this?

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.

16 is the largest number n which has the property of adding n to itself and subtracting 1, then continuing the process by adding half of n and subtraction 1 each iteration to produce primes at each step: 16+16-1 = 31, 16+8-1 = 23, 16+4-1 = 19, 16+2-1 = 17. [Sariyar]

F_16 is the smallest Fermat number greater than five that is also a Self number. [Kontobesar]