# 30

This number is a composite. The largest integer n with the property that every smaller integer relatively prime to n is itself a prime. 3030 + 1 - 30 + 1 is prime. [Luhn] n is a Giuga number if p divides (n/p-1) for every prime divisor p of n. 30 is the smallest such number. 30*2^30-1 is a Woodall prime. [Dobb] 30 is the largest two-digit number such that 30^30+30-1 is prime. [Opao] The product of first five nonzero Fibonacci numbers. Note that 30 + 1 and 30 - 1 are twin primes. [Gupta] Zhi-Wei SUN conjectured in May 2008 that exactly 30 odd integers > 1, all multiples of 3, cannot be written in the form p + n(n+1), where p is a prime congruent to 1 (mod 4) and n a natural number. It is twice more than when p is congruent to 3 (mod 4). [Capelle] Least integer the sum of whose distinct semiprime factors is prime. Semiprimes (6, 10, 15) divide 30, and 6 + 10 + 15 = 31. [Post] The only composite number n such that n^(n+2)+1 is a non-titanic prime, i.e., 30^32+1= 185302018885184100000000000000000000000000000001 (48-digits) is prime. [Loungrides] The smallest sphenic number whose prime factors form a prime-digit prime, i.e., 523. [Loungrides] 30 is the smallest value of n such that n^(n+2)+1 is prime. [Ewing] The smallest integer n such that n^3/(Rn)^3-(Rn) is prime, where Rn is the reversal of n. [Bajpai] The smallest sphenic number, 2*3*5, that can also be represented as sum of two consecutive emirps, i.e., 13+17, is also an oblong number, i.e., 5*6. [Loungrides] The only number n such that reverse of n, n+1 and n+2 is prime. [Gupta]