# 863

This number is a prime. The largest 3-digit prime that can be written as the sum of 15 consecutive primes: 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89. Weintraub discovered that 6505941701960039 is followed by exactly 863 composites. This was the largest prime gap explicitly calculated at the time. 863 = (11 + 71) + (11 * 71). Note that 11, 71 and 863 are primes. [Capelle] sigma(863) = phi(prime(8)*prime(6)*prime(3)). It is the only known multidigit such prime (up to 2*10^8). [Firoozbakht] The smallest prime p in a chain of six consecutive primes {863, 877, 881, 883, 887, 907} such that q = 4*p+9 is also a prime. [Loungrides] Lucas(863) is prime. The largest three-digit prime index for which this is true. [Schiffman] The largest 3-digit prime with distinct digits such that the sum of repdigit numbers formed by repeating d times each of its digits d is prime, i.e., 88888888+666666+333. [Loungrides] There are exactly 863 four-digit odd semiprimes with all distinct digits. [Bajpai]