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 | C_k.
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 {C_k - B_l} > 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.

Fortunate Primes

Conjecture (Original)

A conjecture made by Reo F. Fortune that if $E_k$ = $p_k$# + 1, where $p_k$# is the $k^{th}$ prime and p# is primorial, and assuming $q_k$ is the next prime after $E_k$ (i.e., the smallest prime greater than $E_k$), then $F_k$ = $q_k$ - $E_k$ + 1 is prime for all k. The first values of $F_k$ are 3, 5, 7, 13, 23, 17, 19, 23, ….

Conjecture

Let S₀ = $p_k$# + $p_{k+1}$. If $S_0$ is composite, let $q_0$ be the smallest prime dividing $S_0$. Let $S_i = p_k# + q_{i-1}$, where $q_{i-1}$ is the smallest prime dividing $S_i$.

Assuming $S_0 \lt M^2$, then $S_i \lt {M + 1}^2 \forall$ i. Then $S_n$ is prime for some n ≥ 0. Otherwise, $S_i$ = $S_j$ for some 0 ≤ i < j.

Extended Midy's Theorem

Theorem (Original)

If the period of a repeating decimal for $a/p$, where p is prime and $a/p$ is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. For example, $1/7$ = 0.142857…, and 142 + 857 = 999.

Theorem (Extended)

If the period of a repeating decimal for $a/p$, where p is prime and $a/p$ is a reduced fraction, is h = $m \times n$, then dividing the repeating portion into n parts and adding gives $c_n(a,p)$ × 9 m's, where c is a constant depending on p, a and n.

Click here for my proof.