On consecutive digits

Number Theory Level 3

The identities 12 = 3 × 4 12=3\times 4 and 56 = 7 × 8 56=7\times 8 are solutions of the exponential Diophantine equation 1 0 n l + ( l + 1 ) = 2 m ( 2 m − 1 ) 10^nl+(l+1)=2^m(2^m-1) for n = 1 n=1 .

For which other n n does it have a solution in whole numbers?

4 5 3 2

David Ingerman by David Ingerman , 48, USA

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1 solution

David Ingerman
Jun 22, 2018

Relevant wiki: Exponential Equations - Basic

n = 5 n=5 .

671 × 1 0 5 + 672 = 67100672 = 8191 × 8192 = ( 2 13 − 1 ) × 2 13 671\times 10^5+672=67100672=8191\times 8192=(2^{13}-1)\times 2^{13} .

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