[net.math] Proof wanted

dhc@exodus.UUCP (David H. Copp) (05/01/84)

The author of the previous note requests a proof for the conjecture
"Every positive odd integer >= 3 is the sum of a prime and a power of 2."

The conjecture is false.  The first six counterexamples are
127, 149, 251, 331, 337, 373.
-- 
				David H. Copp

gmf@uvacs.UUCP (05/02/84)

Concerning  x odd & >= 3  -->  x = 2^k + p for some k and prime p .

This may be relevant:

"It can be proved that for every natural number k there are infinitely
many k-powers of natural numbers which are not of the form a^k + p,
where a is an integer and p a prime. (cf. Clement [2])."
                          W. Sierpinski
                          * Elementary Theory of Numbers *
                          (Warsaw, 1964), p. 113

"Clement, P. A. ...
   [2] Representation of integers in the form: a k-th power plus a prime,
       Amer. Math. Monthly 56 (1949), p. 561."
                          Ibid., p. 450

          Gordon Fisher

djc1@ihu1g.UUCP (Dave) (05/07/84)

Can anyone provide me with a proof for the following:

  "Every positive odd integer >= 3 is the sum of a prime and a power of 2."

Please mail me whatever thoughts you may have on solving this.   Thanks....

-- 
		      Dave Christensen
		AT&T Bell Labs - Indian Hill
		     ..ihnp4!ihu1g!djc1