M.I. problem////////\\\\\\\\\

When n = 2 , a2 = 7ᵃ¹ = 7⁷ = 823543 = 20 * 823540 + 3 Assuming that when n = k , ak = 20r + 3 , (r is an integer)
When n = k+1 ,
a k+1
= 7ᵃᵏ
= 7²ºʳ⁺³
= 343 * 7²ºʳ
= 343 * 2401⁵ʳ
= (20*17 + 3) * (20 * 12 + 1)⁵ʳ
= (20*17 + 3) * (20P + 1) , (P is an integer)
= 20 * P(20*17 + 3) + 20*17 + 3
= 20 (P(20*17 + 3) + 17) + 3 is the form of 20r + 3
By M.I. , an can be written as 20r + 3 for n ≥ 2 .
Let am = 20M + 3 , an = 20N + 3 , then
am - an = 20(M - N) is divisible by 20 for m , n ≥ 2 .

2012-04-16 13:51:52 補充：
a₂= 7ᵃ¹ = 7⁷ = 823543 = 20 * 41177 + 3