Question 1137583
<pre>
The above solution is incorrect.

{{{4^503=(2^2)^503=2^1006=2^1006+(2^998+2^990+"..."+2^14+2^6)-(2^998+2^990+"..."+2^14+2^6)=""}}}

      Observe that the sequence of exponents 998,990,... is an arithmetic
      sequence with common difference -8, and smallest positive term 6.
      {{{a[n]=a[1]+(n-1)d=""}}}
      {{{998+(n-1)(-8)=""}}}
      {{{998-8n+8=""}}}
      {{{1006-8n>0}}}
      {{{1006>8n}}}
      {{{125.75>n}}}
      The 125th term is 
      {{{a[125]=998+(125-1)(-8)=6}}}

{{{(2^1006+2^998+"..."+2^14)+2^6-(2^998+2^990+"..."+2^14+2^6)=""}}}

{{{(2^1006+2^998+"..."+2^14)-(2^998+2^990+"..."+2^14+2^6)+2^6=""}}}

{{{2^8(2^998+2^990+"..."+2^6)-(2^998+2^990+"..."+2^14+2^6)+2^6=""}}}

{{{(2^8-1)(2^998+2^990+"..."+2^14+2^6)+2^6=""}}}

{{{(256-1)(2^998+2^990+"..."+2^14+2^6)+64}}}

{{{255(2^998+2^990+"..."+2^14+2^6)+64}}}

So the remainder is 64.

Edwin</pre>