The following are the smallest 9 elements of an infinite set of integers:

0,1,5,6,25,76,376,625,9376

What rule generates the set? What are the next two values?

(In reply to

Answer by K Sengupta)

Each term of the given sequence is an automorphic number. We know that an autiomorphic number is a nonnegative integer T (say) having S digits, such that the last s digits of T^2 when read from left to right constitute the original number, that is T.

For example,

376^2 = __141376__, where 376 is has three digits and the last three digits of 376^2 is indeed 376. Hence, 376 is an automorphic number.

We now observe that the two numbers higher than 9376 with this property are 90625 and 109376, since:

90625^2 = 82128__90625__, and:

109376^2 = 11963__109376__.

Consequently, the required next two terms of the given sequence are __90625__ and __109376__.

*Edited on ***May 19, 2008, 5:08 am**