Maths Item of the Month – April 2012
Think these are true? Don’t be fooled
Find counterexamples to the following conjectures:
1. Every odd integer, N 1, is expressible in the form p + 2n2 where p is prime.
2. If n is an integer, then 991n2 + 1 is not a square number.
3. Polya’s conjecture: For n 1, the number of integers up to and including n with an
odd number of prime factors is never less than the number of integers with an even
number of prime factors.
Solution
Each of these rules are true for so many integers that it is tempting to assume they are true.
However, all of the three do have counter-examples.
Don’t worry if you didn’t find them all: Polya’s conjecture was made in 1919, but the first
counter-example was not found until 1960; 41 years later!
The smallest counter-examples:
1. N = 5777
2. n = 12 055 735 790 331 359 447 442 538 767
3. Even takes the lead when n = 906 150 257
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23/01/13 © MEI