# 73

This number is a prime.

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The smallest prime that divides a 7-digit number of the
form *p*0*p*, where *p* is any 3-digit
prime. [Beedassy]

The largest known prime that starts a chain (generated by
the quadratic 73 - *x* - *x*^{2}, for
*x* = 0, 1, 2, ... , 7) of smaller, increasingly
distant primes with successive gap 2*n*, (*n* =
1, 2, 3, ..., 7): 73, 71, 67, 61, 53, 43, 31, 17. [Beedassy]

The smallest prime with prime digits that belongs both to an emirp pair (37, 73) and to a twin prime pair (71, 73) as the larger member. [Beedassy]

The smallest prime that is the middle term of three
consecutive numbers each expressible as a sum of two
nonzero squares: 72 = 6^{2} + 6^{2} ; 73 =
3^{2} + 8^{2} ; 74 = 5^{2} +
7^{2}. Note that replacing the first prime digit 7
by the prime 23 forms instead the smallest prime (233) that
is the middle term of three consecutive numbers each
expressible as a sum of two *distinct* nonzero
squares: 232 = 6^{2} + 14^{2} ; 233 =
8^{2} + 13^{2} ; 234 = 3^{2} +
15^{2}. [Beedassy]

73 is the only
Sheldon prime, i.e., i) whose binary representation is
palindromic (1001001_{2}) and ii) which belongs to
an emirp pair (P_{n}, P_{m})
such that subscripts (*m*, *n*) = (21, 12) are
also reversals of each other and *n* has a prime
decomposition 21 = 3*7 that concatenates back to
P_{m} (Carl Pomerance, Chris Spicer,
February 2019). [Beedassy]