

A066615


Numbers that are not the sum of two or three distinct primes.


1




OFFSET

1,2


COMMENTS

Probably there are no further terms.
An outgrowth of Goldbach's conjecture. "[I]n a letter to L. Euler (1742), C. F. Goldbach [asserted] that 'every odd number greater than 6 is equal to the sum of three primes.' Euler replied that Goldbach's conjecture was equivalent to the statement that every even number equal to or greater than 4 is equal to the sum of two primes. Because proving the second implies the first, but not the converse, most attention has been focused on the second representation. However, whether the statement is true for all even integers is still unsettled. Nevertheless, it is supported by existing evidence. A Russian mathematician, I. M. Vinogradov, proved that all large odd integers are the sum of three primes. Surprisingly, his techniques involve extremely subtle use of the theory of complex variables; no one has been able to extend them in order to solve Goldbach's conjecture." Andrews.
"Every number greater than 17 is the sum of 3 integers greater than 1 which are relatively prime in pairs."  Wells.


REFERENCES

George E. Andrews, "Number Theory," Dover Publ. Inc., NY, 1994, page 111.
Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, Pages 236257.
Mark Herkommer, "Number Theory, A Programmer's Guide," McGrawHill, NY, 1999, pages 359362.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, "An Introduction to The Theory of Numbers," Fifth Edition, John Wiley & Sons, Inc. NY, 1991, page 2.
Wacław Sierpiński, "250 Problems in Elementary Number Theory," New York: American Elsevier, Warsaw, 1970, pp. 4, 3839.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, 1997, page 76.


LINKS



MATHEMATICA

a = Table[ Prime[n], {n, 1, 100}]; b = {0}; Do[ b = Append[b, a[[i]] + a[[j]]], {j, 2, 100}, {i, 1, j  1}]; Union[b]; c = {0}; Do[ c = Append[c, a[[i]] + a[[j]] + a[[k]]], {k, 3, 100}, {j, 2, k  1}, {i, 1, j  1}]; Union[c]; Complement[ Table[n, {n, 1, 541} ], Union[b, c]]


CROSSREFS



KEYWORD

fini,nonn


AUTHOR



EXTENSIONS



STATUS

approved



