Odd numbers k expressible as 2^n + p, p is prime

$k = 2^{n} + p$,   where   k   is an odd number   $\geq 3$

Here are the first few examples,

$3 = 2^0 + 2$
$5 = 2^1 + 3$
$7 = 2^1 + 5 = 2^2 + 3$
$9 = 2^1 + 7 = 2^2 + 5$
$11 = 2^2 + 7 = 2^3 + 3$
$13 = 2^1 + 11 = 2^3 + 5$
$15 = 2^3 + 7 = 2^2 + 11 = 2^1 + 13$

Can all odd numbers k be expressed as a sum of a power of 2 and a prime p?
Find counterexamples for   $k \leq 1000$

Here’s a counterexample,

$127 = 2^6 + 63 = 2^5 + 95 = 2^4 + 111 = 2^3 + 119 = 2^2 + 123 = 2 + 125 = 2^0 + 126$

Find others.

math grad - Interest: Number theory
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2 Responses to Odd numbers k expressible as 2^n + p, p is prime

1. paul says:

Here are those with k<=1000, refer to the counterexample above for the meaning of the numbers in braces

127 {63,95,111,119,123,125,126}

149 {21,85,117,133,141,145,147,148}
251 {123,187,219,235,243,247,249,250}
331 {75,203,267,299,315,323,327,329,330}
337 {81,209,273,305,321,329,333,335,336}
373 {117,245,309,341,357,365,369,371,372}
509 {253,381,445,477,493,501,505,507,508}
599 {87,343,471,535,567,583,591,595,597,598}
701 {189,445,573,637,669,685,693,697,699,700}
757 {245,501,629,693,725,741,749,753,755,756}
809 {297,553,681,745,777,793,801,805,807,808}
877 {365,621,749,813,845,861,869,873,875,876}
905 {393,649,777,841,873,889,897,901,903,904}
907 {395,651,779,843,875,891,899,903,905,906}
959 {447,703,831,895,927,943,951,955,957,958}
977 {465,721,849,913,945,961,969,973,975,976}
997 {485,741,869,933,965,981,989,993,995,996}

Paul.

• benvitalis says:

Is the set of counterexamples infinite?