Update information on research problem 77

• Book ID: 103058808
• Publisher: Elsevier Science
• Year: 1988
• Tongue: English
• Weight: 153 KB
• Volume: 69
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(88)90184-7

✦ Synopsis

A-ustralia. For any 0 c 9, where 9 is the set of all primes, let &r(a) denote the chromatic number of the graph Z(a) with the integers as vertex-set, and edges between precisely those pairs of integers with absolute differerce in the set 9. It is easy to show that (i) chr(8) = 4, (ii) chr(4)~2if2$~or~={2),and (iii) chr(9)~3if2~~and18~~:2;iffurthermore3$5B,thenchr(~)=3. The problem is to characterize those sets D for which chr@) = 4. . For 9 E 9, &r(9) = 4 precisely when {2,3} c 5B and D coxtains a pair of prime twins. In [l] it is shown that chr@) = 3 if 0: 31 c D and D is a subset of either @ E 9: p = f2 (mod 5)) or (p E 9: p = f2, f3, 7 (mod 14)). It has been proved recently [2] that chr(9) = 4 if {2,3} c 9 and 9 contains a pair of prime tvins; this includes chr(2,3,5) = 4.