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# 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.

30^{30 + 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]