Prime Numbers

434 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
8.13. Prove the claim concerning equation (8.3) under the stated conditions
on k. Start by analyzing the Diophantine equation (mod 4), concluding that
x ≡ 1 (mod 4), continuing on with further analysis (mod 4) until a Legendre
symbol −4m 2
p is encountered for p ≡ 3 (mod 4). (See, for example, [Apostol
1976, Section 9.8].)
8.14. Note that if c = a n + b n ,thenx = ac, y = bc, z = c is a solution to
x n + y n = z n+1 . Show more generally that if gcd(pq, r) = 1, then the Fermat–
Catalan equation x p + y q = z r has infinitely many positive solutions. Why is
this not a disproof of the Fermat–Catalan conjecture? Show that there are no
positive solutions when gcd(p, q, r) ≥ 3. What about the cases gcd(p, q, r) =1
or 2? (The authors do not know the answer to this last question.)
8.15. Fashion an at least somewhat convincing heuristic argument for the
Fermat–Catalan conjecture. For example, here is one for the case that p, q, r
are all at least 4: Let S be the set of fourth and higher powers of positive
integers. Unless there is a cheap reason, as in Exercise 8.14, there should be
no particular tendency for the sum of two members of S to be equal to a
third member of S. Consider the expression a + b − c, wherea ∈ S ∩ [t/2,t],
b ∈ S ∩ [1,t], c ∈ S ∩ [1, 2t] and gcd(a, b) =1.Thisnumbera + b − c is in the
interval (−2t, 2t) and the probability that it is 0 ought to be of magnitude
1/t. Thus, the expected number of solutions to a + b = c for such a, b, c should
be at most S(t) 2 S(2t)/t, whereS(t) is the number of members of S ∩ [1,t].
Now S(t) =O(t 1/4 ), so this expected number is O(t −1/4 ). Now let t run over
powers of 2, getting that the total number of solutions is expected to be just
O(1).
8.16. As in Exercise 8.15, fashion an at least somewhat convincing heuristic
argument for the ABC conjecture.
8.17. Show that the ABC conjecture is false with ɛ = 0. In fact, show
that there are infinitely many coprime triples a, b, c of positive integers with
a + b = c and γ(abc) =o(c). (As before, γ(n) is the largest squarefree divisor
of n.)
8.18. [Tijdeman] Show that the ABC conjecture implies the Fermat–Catalan
conjecture.
8.19. [Silverman] Show that the ABC conjecture implies that there are
infinitely many primes p that are not Wieferich primes.
8.20. Say q1 0, we have
qn+1 − qn >n 1/12−ɛ for all sufficiently large values of n.
8.21. Show that there is a polynomial in two variables with integer
coefficients whose values at positive integral arguments coincide with the set