James Propp <propp@math.wisc.edu> wrote on 4 Jun 2003:
When you write the numbers from 1 to n^2 in n rows of length n (with n > 1 arbitrary), does each row contain a prime?
This was conjectured by Sierpinski in his famous 1958 joint paper with Schinzel which presented Schinzel's Hypothesis H. You'll find much about these and related conjectures in Ribenboim's: The New Book of Prime Number Records, Chapter 6. It seems this conjecture has little presence on the web - a simple Google search turned up only one hit on Carlos Rivera's Prime Puzzles site, excerpted below: ------------ Excerpt from: http://www.primepuzzles.net/conjectures/conj_026.htm CONJECTURE 26. The Calendar-like square Conjecture Julio Cesar Aguilar, from Mexico, has the following double-conjecture: In a calendar-like square array of numbers from 1 to p^2, p prime: a) there is always at least one prime per row b) there is always at least one prime per column For example if p=5, this is the calendar-like square and the primes inside: 1 *2 *3 4 *5 6 *7 8 9 10 *11 12 *13 14 15 16 *17 18 *19 20 21 22 *23 24 25 QUESTIONS: 1. Can you find a counterexample? 2. Otherwise, what can you argue on favor of each part of the conjecture? 3. How is this conjecture related to other known Conjectures? SOLUTION Luis Rodriguez wrote (17/11/01): "In reference to the rows, this conjecture is equivalent to the Sierpinski conjecture (Ribenboim p. 397). The Sierpinski conjecture can be expressed more succinctly: 'For any n >= 2 and any 1 < k <= n there exists a prime number between (k-1)n and kn.'" About Sierpinski's Conjecture (S) Ribenboim's book says: "(S) For every integer n>1, let n^2 integers 1, 2, 3, ... n^2 be written in an array with n rows, each with n integers, like an n x n matrix: 1 2 ... n n+1, n+2 2n . . . . . . . . . (n-1)n+1 (n-1)n+2 ... nn Then, there exists a prime in each row" Moreover, I found two pages ahead, that Schinzel made the "transposed form of the Sierpinski Conjecture", or the second part of Aguilar's Conjecture: "(S') For every integer n>1, let n^2 integers 1, 2, 3, ... n^2 be written in an array with n rows, each with n integers (just like S). If 1 <= k <= n and gcd(k,n) = 1, then the kth column contains at least one prime" Schinzel & Sierpinski wrote these Conjectures in 1958. According to Ribenboim they added "We do not know what will be the fate of our hypotheses, however we think that, even if they are refutes, this will not be without profit for number theory". Ribenboim comments "It's my impression that none of the conventional present day methods of number theory will lead to a proof of any of the conjectures (D), (B), (H), (S), (S'). Perhaps will be the role of the logicians, investigating the inner structure of arithmetic, to decide whether such statements are, or are not, provable from the Peano axioms." Jud McCranie wrote "I tested it for p <= 23929, and it holds for those values. It is almost certainly true". Aguilar has tested the column-conjecture for p < 790000. Ribenboim adds that "In 1963 conjecture (S') was spelled out once more by Kanold...(and) a quantitative version of the Conjectures in the paper of Schinzel and Sierpinski was formulated by Bateman and Horn (1962)" [...] Follow the above link for the rest of the article. -Bill Dubuque