# 34

This number is a composite. 34 = π(144). Note that 34 and 144 are Fibonacci numbers. No larger example of this type is known. [Honaker] The first 34 odd numbers (concatenated) is prime. [Das] 34 is the smallest number which can be expressed as the sum of two primes in four ways. [Murthy] The smallest composite Fibonacci number whose sum of prime factors is a prime. [Gupta] 134 + 233 + 332 + ... + 323 + 332 + 341 is prime. [Patterson] 34 = 3 + 5 + 7 + 19 is the smallest number which is sum of distinct primes whose digits are odd. Note that all the odd digits are used, without repetition. [Capelle] The smallest Fibonacci number f such that neither 6f - 1 nor 6f + 1 are prime. [Necula] π(34)= !3 + !4, where !3 and !4 denotes subfactorial 3 and subfactorial 4 respectively. [Gupta] 34 = π(3!*4!). [Firoozbakht] 34!/34# ± 1 are twin primes. [Wesolowski] π(34) = 3!! + 4!!. [Gupta] There are 34 five-digit primes formed from the five odd digits. This means there's a Fibonacci number of Fibonacci-digit primes formed from the Fibonacci number of odd digits. [Silva and Honaker] 7^34+34 is the smaller of only two non-titanic primes of form 7^n+n, (the other is for n=48). [Loungrides] 34 is a Fibonacci number F(9) that is simultaneously the sum of the squares of two consecutive primes (F(9) = 34 = 3^2 + 5^2) and the sum of the squares of two consecutive Fibonnaci's (F(9) = 34 = F(4)^2 + F(5)^2). [Rivera] The number of the distinct-digit primes each consisting of all of the odd digits. These are: 13597, 13759, 15739, 15937, 15973, 17359, 17539, 19753, 31957, 37159, 37591, 37951, 39157, 51973, 53197, 53719, 53791, 53917, 57139, 57193, 71359, 71593, 73951, 75193, 75391, 75913, 75931, 79153, 79531, 91573, 91753, 95317, 95713, 95731. [Loungrides] The smallest semiprime that is the sum of two consecutive prime squares (2*17=34=3^2+5^2). [Bajpai] First occurrence of a run of exactly 34 consecutive integers with an odd number of prime factors has never been found. Along with 34+1, the smallest pair of consecutive integers that are both composite and reverse primes. [Gaydos] Of all the primes produced by 34n + n - (n + 1)² for n = 1 - 34, only one is an isolated prime. Although the twin primes produced are not distinct it is interesting to note that there are two twin primes in the first three n values, three from 5, i.e., 5/8, 8/13, 13/21 and twenty-one twin primes from thirty-four n values, giving a ratio of primes produced exactly equal to the Golden Ratio throughout the sequence. [Homewood]

(There are 6 curios for this number that have not yet been approved by an editor.)