In Mathematics
101 is the 26th prime number and a palindromic number (and so a palindromic prime). The next prime is 103, with which it makes a twin prime pair, making 101 a Chen prime. Because the period length of its reciprocal is unique among primes, 101 is a unique prime. 101 is an Eisenstein prime with no imaginary part and real part of the form .
101 is the sum of five consecutive primes (13 + 17 + 19 + 23 + 29). Given 101, the Mertens function returns 0. 101 is the fifth alternating factorial.
101 is a centered decagonal number.
The decimal representation of 2^101-1 is 2 535301 200456 458802 993406 410751. The prime decomposition of that is 7 432339 208719 x 341117 531003 194129
For a 3-digit number in base 10, this number has a relatively simple divisibility test. The candidate number is split into groups of four, starting with the rightmost four, and added up to produce a 4-digit number. If this 4-digit number is of the form 1000a + 100b + 10a + b (where a and b are integers from 0 to 9), such as 3232 or 9797, or of the form 100b + b, such as 707 and 808, then the number is divisible by 101. This might not be as simple as the divisibility tests for numbers like 3 or 5, and it might not be terribly practical, but it is simpler than the divisibility tests for other 3-digit numbers.
On the seven-segment display of a calculator, 101 is both a strobogrammatic prime and a dihedral prime.
101 is the only existing prime with alternating 1s and 0s in base 10 and the largest known prime of the form 10n + 1.
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