



 
Hi Peter, Suppose the problem was to factor x^{2}  729. This I can do since 729 = 27^{2} and thus x^{2}  729 = x^{2}  27^{2} is a difference of squares and thus
In particular this means that if x is an integer then the integer x^{2}  729 factors. For example if x = 2 then
Now what about x^{2} + 729? In this case if x = 2 then x^{2} + 729 = 733 which is a prime and hence doesn't factor except as 1 × 733. Thus the only possible factorization of x^{2} + 729 is 1 × (x^{2} + 729) and hence we say that x^{2} + 729 doesn't factor. Harley Peter wrote back
Since 729 = 9^{3}, if you write z = x^{2} then
which is a sum of cubes. Sums of cubes do factor nicely
Harley  


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