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It suffices to find a counterexample. After some searching, we find A133122 Odd numbers which cannot be written as the sum of an odd prime and a power of ...
Odd numbers not of form p + 2^k (de Polignac numbers). (Formerly M5390), 29. 1, 127, 149, 251, ...
OFFSET. 1,2. COMMENTS. The sequence of "obstinate numbers", that is, odd numbers which cannot be written as prime + 2^i with i >= 0 is the same except for ...
May 24, 2017 · The sequence of obstinate numbers begins: 1,127,149,251,331, ...
May 24, 2017 · Sequence of Obstinate Numbers. The sequence of obstinate numbers begins: 1,127,149,251,331,337,373,509,599,701,757,809,877,905,…
prime. The name comes from the fact that de Polignac erroneously conjectured that every odd number can be expressed in that way. Erdös proved that there are ...
References to A065381. Primes not of the form p + 2^k, p prime and k >= 0. 2, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 907, 977, 997, 1019 ...
36525 matches · 2, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 907, 977, 997, 1019, 1087, 1259, 1549, 1597, 1619, 1657, 1759, 1777, 1783, ...
{1=f ⍵:⍵⋄⍬}¨1..1000 1 127 149 251 331 337 373 509 599 701 757 809 877 905 907 959 977 997. for input <=0 or input >=9E9 it return ¯1 (error)
A brief spreadsheet search on the primes gives 149 as a counterexample. 149−20=148, even 149−21=147=3⋅72 149−22=145=5⋅29 149−23=141=3⋅47