1+2+2^2+...+2^219=2^(220)-1. Now 2 has order 12 mod 13; since 220 is 4 mod 12, 2^220 is 2^4 mod 13 or 3. So then your integer is 2 mod 13.
Hello, dave52!
We have: .
Consider consecutive powers-of-2 (mod 13).
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We see that: .
We have: .
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