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You can get away with only using work of Chebyshev for large enough a: Let f(n) = \sum log(p) over all primes p up to n (usually denoted theta(n)). If a+b<c then c>2a and so there's at most one prime between a+1 and 2a, hence f(2a)-f(a) < log(2a). He showed that f(a) < a*log(4), and he proved a bound pi(N) > 0.9N/log(N) for N large, so we should have f(a) >= 0.7a for a large. Then for such a we have log(2a) > f(2a)-f(a) >= (1.4-log(4))a > 0.0137a, which is impossible if a is large in the above sense and at least 505.
@Agol: Monopole Floer homology is another name for SW Floer homology, and in any case the two sutured Floer homologies are in fact isomorphic as abelian groups (i.e. maybe without a decomposition with respect to spin^c structures). This follows from work of Lekili (arxiv.org/abs/0903.1773) together with the equivalence of Heegaard Floer and monopole Floer homologies proved by either Kutluhan-Lee-Taubes or Taubes + Colin-Ghiggini-Honda.
Knots do decompose uniquely into a sum of primes, see e.g. chapter 2 of Lickorish's "An introduction to knot theory." Also, since knot genus is additive under connected sum it follows that every genus 1 knot is prime, so take your favorite knot and consider all of its twisted Whitehead doubles; these have genus 1 and are distinguished by their Alexander polynomials.
Dynnikov's paper "Arc-presentations of links. Monotonic simplification" (arXiv:0208153) was mentioned several times in answers to the unknot recognition question. The algorithm in that paper can also recognize split links and hence unlinks, and it does so without ever increasing the size of the diagram, but I don't think there are any good (e.g. subexponential) upper bounds known on the number of moves it requires.
@Dan: Bizaca's paper "An explicit family of exotic Casson handles" shows that some Casson handles are exotic using the fact that iterated Whitehead doubles of the trefoil are not smoothly slice; at the time this required gauge theory, but it has since been done by Hedden ("Knot Floer homology of Whitehead doubles") using the Ozsváth-Szabó tau invariant.
Since the homology groups of a closed 3-manifold are determined by the fundamental group, any pair of non-homotopy-equivalent 3-manifolds with the same homotopy groups should work, like the lens spaces L(5,1) and L(5,2) I mentioned in a comment at the above link.
