A question regarding the method followed in Cohen & Selfridge's paper on covering systems. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:30:52Z http://mathoverflow.net/feeds/question/101297 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101297/a-question-regarding-the-method-followed-in-cohen-selfridges-paper-on-covering A question regarding the method followed in Cohen & Selfridge's paper on covering systems. Nikhil Bellarykar 2012-07-04T09:23:01Z 2012-07-04T09:38:47Z <p>I am reading this paper by Cohen and Selfridge that deals with covering systems. Its link is</p> <p><a href="http://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf" rel="nofollow">http://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf</a></p> <p>Here, they demonstrate the existence of a number that is not the sum or difference of two primes powers; using covering systems.</p> <p>First, they construct the covering systems and obtain some congruences that demonstrate that the given number is not the sum or difference of a power of 2 and a prime. This I understand without any difficulty.</p> <p>Then, for proving that the number such obtained is not the sum of a power of two and a POWER of prime,they introduce some extra congruences. The basic idea is that if one could show the reqd. number to be multiple of a product of two or more primes, then it won't be a prime power.</p> <p>That is fine, but take a look at the primes in extra congruences. From where are these extra primes selected? e.g. on page 2 of the paper, we have:</p> <p>$M + 2^8 \equiv 0\pmod{5^2 \times 11}$, $M + 2^{34} \equiv 0\pmod{97^2 \times 389}$, $M + 2^6 \equiv 0 \pmod{17^2\times 137}$, $M + 2^{18} \equiv 0\pmod{241^2\times 1447}$ $M+ 2^2 \equiv 0 \pmod{13^2 \times 53}$ $M + 2^{10} \equiv 0 \pmod{257\times 673}$</p> <p>In the above congruences, the primes 5,97,17,241,13,257 all come from the covering (1,2), (0,4), (6,8), (10,12), (2,48), (10,16), (18,24).</p> <p>But where do the primes 11, 389, 137, 1447, 53, 673 etc. come? The same question I have regarding the extra congruences for the other covering given in the paper.</p> <p>The authors have given no reasoning for the same in the paper.</p> <p>I checked the prime factors of $2^{48}-1$ &amp; $2^{180}-1$ as well. The additional primes in both cases are different from those.</p> <p>Hence the question: what is the rationale behind selecting the primes in the extra congruences?</p>