Jev's favorite number
142857. Its multiples rotate its digits. Its square splits back into itself. Both patterns have short proofs. 142857 × 2 = 285714; × 3 = 428571; × 4 = 571428; × 5 = 714285; × 6 = 857142; × 7 = 999999. 142857² = 20408122449, and 20408 + 122449 = 142857. The rotations are one structure, the return through the square another. Powers of ten modulo seven explain the first; N²≡N modulo 999999 explains the second. The patterns are exact, not rounded coincidences. It is a little piece of order that stays surprising even after you know how it works.
142857. A pocket-sized demonstration that numbers have structure beyond their size. 142857 × 2 = 285714; × 3 = 428571; × 4 = 571428; × 5 = 714285; × 6 = 857142; × 7 = 999999. 142857 squared is 20408122449. Split it into 20408 and 122449, add them, and you recover 142857.
142857. A pocket-sized demonstration that numbers have structure beyond their size. 142857 × 2 = 285714; × 3 = 428571; × 4 = 571428; × 5 = 714285; × 6 = 857142; × 7 = 999999. 142857 squared is 20408122449. Split it into 20408 and 122449, add them, and you recover 142857. The reason for the rotations is simple: 10 has order six modulo 7. Its successive powers leave remainders 1, 3, 2, 6, 4, 5. Long division follows that loop, producing 142857. Starting from a different nonzero remainder changes only where in the loop the digits begin. It is a little piece of order that stays surprising even after you know how it works.
6174 7641 − 1467 = 6174. 3524 → 3087 → 8352 → 6174 → 6174. At each step, arrange the digits descending and ascending, then subtract. If a≥b≥c≥d are the sorted digits, the subtraction is 999(a−d)+90(b−c). Thousands of starting strings collapse to a few digit differences, which explains why the process can have a small destination. It is a little piece of order that stays surprising even after you know how it works.
65537. A pocket-sized demonstration that numbers have structure beyond their size. 65537 = 2^16+1. It is prime. 65536 = 2^16, so the multiplicative group of nonzero residues modulo 65537 has power-of-two order. A regular 65537-sided polygon is constructible with straightedge and compass. It is a little piece of order that stays surprising even after you know how it works.
Every score is Jev’s own opinion. 0 is the least favorite; 1,000 is the absolute favorite.