It's a Puzzlementby John P. Robertson
Unsolved Problems in Number Theory
There are thousands of unsolved problems in number theory. One famous open problem is due to Goldbach: prove or disprove that every even integer greater than 2 is a sum of two prime numbers. A less well-known problem is that of Egyptian fractions: determine whether every fraction of the form 4/n with n > 1 can be written as a sum of three positive rational numbers with numerator 1, i.e. 4/n = 1/i + 1/j + 1/k.
Despite the fact that there are so many unsolved problems, Gary Venter continues to offer new problems. Here are two. The first is to investigate sequences of consecutive positive integers where each term is a product of two or more two-digit numbers (not necessarily primes). For example, 322, 323, 324, 325 give four such consecutive integers (322=14x23, 323=17x19, 324=18x18, and 325=13x25). What is the longest sequence you can find?
In his second problem, Gary noted that if n=12 (or 36, or 156) then (n+1)/1, (n+2)/2, and (n+3)/3 are all prime. What is the largest k you can find such that there is an n with (n+1)/1, (n+2)/2, ... , (n+k)/k all prime?
Does anyone know whether there is any theory that might suggest what to expect for either of these problems? For an integer n in the range of one million to one billion it appears, empirically, that the probability that n is a product of two-digit numbers is roughly 22n-0.45. Assuming that the probabilities of n and n+1 being expressible as products of two-digit numbers are statistically independent, this suggests that the expected number of sequences of length 3 or greater in any range [n, a'n], for a fixed a, declines as n gets larger. For the (n+i)/i problem, I don't see any reason why there should not be arbitrarily large k for which there is an n with the required properties. Does anyone have any thoughts on this?
Babies - A Pink or Blue Addition Problem
The problem had a nurse counting the babies in a hospital nursery. He has just counted two boys, and has not counted the girls, when, at 11:00, a new baby is brought to the nursery. A baby is then selected at random, from all the babies present, to have its footprint taken. The selected baby happens to be a boy. What is the probability that the baby added at 11:00 was a girl?
Melissa Neidlinger applied Bayes' Theorem, letting x be the starting number of girls and 2 the starting number of boys. She noted that if a boy arrived at 11:00 then the final number of boys is 3, and the final number of girls is x, while if a girl arrived, then these are 2 and x+1 respectively. Either way, the final number of babies is x+3. She assumed that the unconditional probability that the added baby was a girl was 50%. The probability that the added baby was a girl is 40 percent. For a copy of the formula used to solve this problem, contact the CAS Office.
Solutions were also sent in by Alex Bondarev, Mary Ellen Cardascia, Costas Constantinou, Steve Darrow, Jennifer Grimes, Kevin Kelso, Daniel Kligman, Bob Montgomery, John Noble, Randy Nordquist, Steve Philbrick, Nathan Schwartz, Michael Shackleford, David Skurnick, Russell Wenitsky, and jointly by William Finn and Sak Man Luk. Solutions arrived by regular mail, fax, and Internet E-mail. Michael Shackleford has a Web page with additional puzzles at http://www.charm.net/~shack/math/.