Suat AyözProblem for May 6th from the 2026 AMS Daily Epsilon of Math Calendar
Alex Bolton@PercyButtons3 @DailyEpsilon We can see that the gcd can't be bigger than 6 by checking the first two terms.
a_1 = 2028 = 2² × 3 × 13²
a_2 = 4,104,678 = 2 × 3 × 684,113 (684,113 is prime)
But does 6 divide all the terms? 2026 ≡ 4 (mod 6) and 4² ≡ 4 (mod 6) so we have 2026ⁿ ≡ 4ⁿ ≡ 4 (mod 6) for n = 1, 2, .... Thus 2026ⁿ + 2 ≡ 0 (mod 6) for any n = 1, 2, ...
a_1 = 2028 = 2² × 3 × 13²
a_2 = 4,104,678 = 2 × 3 × 684,113 (684,113 is prime)
But does 6 divide all the terms? 2026 ≡ 4 (mod 6) and 4² ≡ 4 (mod 6) so we have 2026ⁿ ≡ 4ⁿ ≡ 4 (mod 6) for n = 1, 2, .... Thus 2026ⁿ + 2 ≡ 0 (mod 6) for any n = 1, 2, ...