Readers interested in this topic should probably read Terrence Tao’s excellent discussions of probabilistic heuristics in number theory, e.g. this post discussing Fermat’s last theorem, the ABC conjecture, and twin primes or this post on biases in prime number gaps. Those posts really helped improve my understanding of how such heuristic arguments work, and there’s some cool surprises.
Isn’t the Stuart conjecture an extremely weak form of the Lander, Parkin and Selfridge conjecture? If you specialize their conjecture to m=1 then it implies your conjecture but with k≥n−1 instead of k≥n/2. (I’m not sure why you picked n/2).
I think the fundamental thing about Fermat’s last theorem for n=3 is that you can divide by any common divisor of x and y. Once you consider a minimal solution with (x, y, z) relatively prime, and consider the factorization x3+y3=(x+y)(x2−xy+y2), the n=3 case is also very straightforward to argue heuristically. [ETA: this is not true, it actually seems to fundamentally be a subtle claim related to the elliptic curve structure.]
Readers interested in this topic should probably read Terrence Tao’s excellent discussions of probabilistic heuristics in number theory, e.g. this post discussing Fermat’s last theorem, the ABC conjecture, and twin primes or this post on biases in prime number gaps. Those posts really helped improve my understanding of how such heuristic arguments work, and there’s some cool surprises.
Isn’t the Stuart conjecture an extremely weak form of the Lander, Parkin and Selfridge conjecture? If you specialize their conjecture to m=1 then it implies your conjecture but with k≥n−1 instead of k≥n/2. (I’m not sure why you picked n/2).
I think the fundamental thing about Fermat’s last theorem for n=3 is that you can divide by any common divisor of x and y. Once you consider a minimal solution with (x, y, z) relatively prime, and consider the factorization x3+y3=(x+y)(x2−xy+y2), the n=3 case is also very straightforward to argue heuristically. [ETA: this is not true, it actually seems to fundamentally be a subtle claim related to the elliptic curve structure.]
Because it’s the first case I thought of where the probability numbers work out, and I just needed one example to round off the post :-)