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.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

math grad - Interest: Number theory
This entry was posted in Prime Numbers and tagged . Bookmark the permalink.

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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s