# 10

This number is a composite. 10 = 2 + 3 + 5 (sum of the first 3 primes). The smallest unresolved non-prime order to Lam's Problem. No number of even 10 digits can be a no-rep emirp. [Dale] 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 + 1010 is prime. [Kulsha] There exists a polynomial in 10 variables with integer coefficients such that the set of primes equals the set of positive values of this polynomial obtained as the variables run through all nonnegative integers. [Ribenboim] If the number of the month and the number of the day are relatively prime, then June has the fewest such days ... 10. The smallest number which is the sum of two primes in two ways (10 = 3 + 7 = 5 + 5). There are 10 non-twin primes less than 102. [Francis] The 10-bit number (1110101001) containing zeros on prime positions and units on non-prime positions, reading right-to-left, is a binary emirp. [Kulsha] 10! - 9! + 8! - 7! + 6! - 5! + 4! - 3! + 2! - 1! is prime. [Poo Sung] The smallest composite number (the fifth one) which is smaller than the corresponding (the fifth) prime 11. [Murthy] The only number with the property that the sum as well as the difference of its prime divisors are primes (2 + 5 = 7 and 5 - 2 = 3). [Murthy] (10!)2 + 1 is prime. [Dobb] The Top 10 is a catalogue of primal configurations by Rudolf Ondrejka. The smallest even number whose square is the sum of n first primes. [Gallardo] 10! = 3! 5! 7! Note the first three consecutive odd primes. [Capelle] 10 ones plus 10 is prime. This is the smallest such two-digit number. [Opao] 10! + 11 and 10! - 11 are primes. [Patterson] 10!+9!+8!+ ... +2!+1!+0!+1!+2!+ ... +8!+9!+10! is prime. [Silva] 10 is the only number n such that n^2 = prime(2*n) + prime(n). [Firoozbakht] The largest number N such that any set of N consecutive integers contains at least one integer relatively prime to all other integers in the set. [Rupinski] π(10) = prime(1!) + prime(0!) [Firoozbakht] In the year '10 of this (or any) century, a person born in '73 of the previous century will be 37 and a person born in '37 will be 73. This pattern is true of many different years and ages, but this is the only pattern represented by two emirps. [Green] The number of known perfect powers n with no primes between n and the next larger perfect power. [Capelle] The smallest reflectable semiprime and also the smallest reflectable brilliant number. [Capelle] 10+98765432123456789+10 is prime. [Silva] 10^10+9^10+8^10+7^10+6^10+5^10+4^10+3^10+2^10+1^10+2^10+3^10+4^10+5^10+6^10+7^10+8^10+9^10+10^10 is an 11-digit prime. [Silva] The smallest integer as a prime number of knots with cyclic symmetry group Z_n for prime n, is knots with 10 crossings, of which 3 have symmetry Z_2. [Post] The number of palindromic primes that can be displayed on a digital clock (both 12-hour and 24-hour displays). [Fellows] The smallest integer that is not a prime power and not covered by the Bruck-Ryser theorem. 10 question quiz on prime numbers. The only known number n such that the forms n+R(n), n^2+R(n^2), 3^n+2, 3^(R(n))+2 are all primes, where R(n) is the reversal of n, i.e., 10+01=11, 100+001=101, 3^10+2=59051, 3^01+2=5. [Loungrides] The only even number which can be expressed as the sum of three consecutive primes. [Silva] 10 is the only 1 semiprime with end-digit 0. [Silva] The 10 prime numbers 1103, 1109, 1117, 1667, 1877, 3001, 3389, 3559, 4001, and 4517 each has the same Collatz trajectory length as its prime index. There are no more of these in the first billion primes. Are there any more at all? [Gaydos] The sum of the first two primes of form 4k + 3, i.e., 3 + 7 = 10. [Rivera] The only known even semiprime that is the sum of the primes between its two prime factors (10 = 2 * 5 = 2 + 3 + 5). [Gupta]

(There are 29 curios for this number that have not yet been approved by an editor.)