## Thursday, July 28, 2005

### ABC Conjecture

Conjecture

The ABC conjecture is a conjecture due to Oesterlé and Masser in 1985. For any three relatively prime numbers A, B, and C, such that A + B = C, and any infinitesimal $\epsilon$ > 0, there exists a constant c$\epsilon$ such that $sqp(ABC)^{1 + \epsilon} \ge c_{\epsilon} \times C$, where sqp(n) is the square-free product of all primes dividing n.

Modified ABC Conjecture

Given a fixed set of distinct primes $p_i$ and $q_j$.
Let C = $\prod_{i=1}^n p_i, B = \prod_{j=1}^m q_j$, where ($p_i$, $q_j$) = 1.
Let O(C) = {$C_k$} = {$\prod p_i^{c_i}, c_i \ge 1$}, be the infinite ordered set of all numbers such that C | Ck.
Let O(B) = {$B_l$} = {$\prod q_j^{b_j}, b_j \ge 1$}, be the infinite ordered set of all numbers such that B | $B_l$.
Let $A_k$ = min {Ck - Bl} > 0, ∀ l.
Let $k^'$ = {k | $A_{k^'}$ = inf($A_k$)}.

Then, $k^'$ > 0 and sqp($A_k B_l C_k$) = $B C \times sqp(A_k) \ge C_k \times \epsilon$, where $\epsilon = B C \times sqp(A_{k^'})$ / $C_(k^'}$.

The original ABC conjecture states that for a given $\epsilon$, there is a constant $c_\epsilon$ that is based only on $\epsilon$ such that the inequality holds.

The modified conjecture states that for fixed distinct primes dividing B and C, we can find a constant $c_{p,q}$ that depends only on p and q that also satisfies the inequality.