631

This number is a prime.

One of the record-breaking differences between consecutive Ulam numbers (332250401 and 332251032). [McCranie]

631 values of n ≤ 1000 produce primes for 2n² + 144251. Note that 631 is prime also. [McLean]

The reverse concatenation of the first three triangular numbers. [Gupta]

The tetractys (pronounced "tet-trak'tis") is a triangular figure consisting of ten vertices arranged in four rows: one, two, three, and four dots in each row. It was a mystical symbol to the Pythagoreans, who lived during the 6th century B.C. There are fifteen primes in the figure below (reading forwards or backwards along the indicated lines), the largest of which is 631.

                    (0)
                    / \
                  (1)-(2)
                  / \ / \
                (3)-(4)-(5)
                / \ / \ / \
              (6)-(7)-(8)-(9)

Can you rearrange these digits and achieve two dozen primes?

The constant 10πe³ is very close to 631. [Leonardis]

The largest in the fist case of five consecutive primes of form 5^n+6 for n= 0, 1, 2, 3, 4. The sequence of these primes is: 7, 11, 31, 131, 631. [Loungrides]

The reversal of the sum of squares of two consecutive triangular numbers (T₃ and T₄). The smallest prime of this form. [Bajpai]

The smallest prime p such that p^6 starts with p (631^6 = 63121332085847281). [Gupta]