# Number Gossip

(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)

## Unique Properties of 31

• 31 is the earliest and the only known case such that the sum of the divisors of two distinct numbers (16 & 25) is the same prime quantity (31), that is to say: 1+2+4+8+16 = 31 and 1+5+25 = 31
• Given six points on a circle connected by lines, the maximum number of regions is 31. This answer is famous for breaking a pattern as if there were fewer points, the number of regions would be a power of 2
• 31 is the number of minimal composites which cover the set of composites in base 10
• There are only 31 numbers which cannot be expressed as the sum of distinct squares
• There are only 31 numbers which when multiplied by any power of 4 are not the sums of 4 distinct squares
• 31 = 22 + 33, i.e., the sum of the first two primes raised to themselves
• 31 is the smallest prime that is a sum of two triangular numbers with prime indices
• 31 is the smallest prime p such that neither 6p-1 nor 6p+1 are primes