(a) Find the form of all positive integers satisfying What is the smallest positive integer for which this is true? (b) Show that there are no positive integers satisfying .
Step 11 of 3
- Let be a positive integer satisfying . If , then
Since has exactly two prime factors, we see that can have at most two prime factors (else it would have too many factors). From this we have the two following possibilities for :
for prime . The smallest number for which of the first form is and of the second form is . It is readily seen that the other number is smaller, thus is the smallest number such that .