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ABC Conjecture

We define the radical R(n) of an integer n to be the the product of its distinct prime divisors.

ABC Conjecture: For every k>1, there is a constant C_k such that if a,b,c are coprime positive integers satisfying a+b=c, then c < C_k R(abc)^k.

The conjecture was proposed by Osterle and Masser in 1985. At present the best that can be proved is that c < exp(R(abc)^f) for a suitable f.
Mason has proved the analogue of the conjecture for polynomials.

If true, the conjecture would have numerous important consequences: among them would be another proof of Fermat's Last Theorem.

We define the ABC ratio for a triple (a,b,c) to be A = log(c) / log(R(abc)). The conjecture implies that A is bounded, so it is of interest to find large values of A.
The current best is: a=2, b=3¹⁰.109, c=23⁵ giving A = 1.6299

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Collatz Problem

Also known as the 3n+1 problem, Syracuse problem, Thwaites problem.
Define a function on positive integers n -> 3n+1 if n is odd, n/2 is n is even. Does every positive integer eventually end up in the cycle 4 -> 2 -> 1 -> 4?

It is known that there is no other cycle of length less than 275,000 and that all numbers up to 198.2⁵⁰ eventually "converge" to 1.

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Goldbach Conjecture

Every even number except 2 is the sum of two primes, proposed by Christian Goldbach in correspondence with L. Euler in 1742.

The conjecture has been numerically verified up to 2.10¹⁶.
Vinogradov proved that every sufficiently large odd number is the sum of three primes, and Chen proved in 1973 that every sufficiently large even number is the sum of a prime and a number with at most two prime factors.