A Halmos Problem and a Related Problem

Abstract

I first saw this problem in a British Sunday newspaper, the Sunday Times, about fifteen years ago I don't know who first thought of it, perhaps some British reader could provide a reference but only last October did I.present it, for the first time to any of my students, to my first year students, who are training to be primary school teachers. Then I was introducing them to Number Theory and had asked them to find some primes which are a power of 2 plus 1. They quickly came up with 3, 5 and 17 which are (21 + 1), (22 + 1) and (24 + 1), and one of them said 'there's a pattern!' Good! Namely? 'The next one will be (28 + 1).' They checked that it is prime, and I asked how could they be sure that it is the next prime of this type. They tested all the 'missing' ones, (23 + 1), (25 + 1), (26 + 1) and (27 + 1), all composite. <[Later I showed, for example, (235 + 1) can be seen to be composite: (235 + 1) = (25 + 1)(23°-225 + 22o_ 215 + 21°-25 + 1)] And the next one? I was of course, given (216 + 1); and then? Well I was told the next one is (232 + 1). It was then I let out that they had just rediscovered Fermat's false claim with respect to his primes (that the 'Fermat numbers' Fn (= 22 + 1) are prime for all n = 0, 1, 2,3, . . . ) and got them to verify that 641 divides (232 + 1) (Euler). I also asked them to find primes that are a power of 2 less 1; here they quickly found 3, 7, 31 and 127 which are (22 1), (23 1), (251) and (271). Again someone saw a 'pattern,' claiming the next to be (21l 1), but it is composite, being 23.89 (first noted in the 15th century). Then I presented this problem (not the 'related' one of the title): Some natural numbers can/can't be expressed as a sum of at least two consecutive natural numbers; find some of each kind. Check down to 20. Soon they had two lists: