A Survey on the Ternary Purely Exponential Diophantine Equation ax+by=cza^x + b^y = c^z

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$

At most one solution to ax+by=cza^x + b^y = c^z for some ranges of aa, bb, cc

We consider the number of solutions in positive integers $(x,y,z)$ for the purely exponential Diophantine equation $a^x+b^y =c^z$ (with $\gcd(a,b)=1$). Apart from a list of known exceptions, a conjecture published in 2016 claims that this equation has at most one solution in positive integers $x$, $y$, and $z$. We show that this is true for some ranges of $a$, $b$, $c$, for instance, when $1 < a,b < 3600$ and $c<10^{10}$. The conjecture also holds for small pairs $(a,b)$ independent of $c$, where $2 \le a,b \le 10$ with $\gcd(a,b)=1$. We show that the Pillai equation $a^x - b^y = r > 0$ has at most one solution (with a known list of exceptions) when $2 \le a,b \le 3600$. Finally, the primitive case of the Je\'smanowicz conjecture holds when $a \le 10^6$ or when $b \le 10^6$. This work highlights the power of some ideas of Miyazaki and Pink and the usefulness of a theorem by Scott

Number of solutions to ax+by=cza^x + b^y = c^z, A Shorter Version

For relatively prime integers $a$ and $b$ both greater than one and odd integer $c$, there are at most two solutions in positive integers $(x,y,z)$ to the equation $a^x + b^y = c^z$. There are an infinite number of $(a,b,c)$ giving exactly two solutions.Comment: added explanatory comments, fixed minor error

Two terms with known prime divisors adding to a power: REVISED

Let $c$ be a positive odd integer and $R$ a set of $n$ primes coprime with $c$. We consider equations $X + Y = c^z$ in three integer unknowns $X$, $Y$, $z$, where $z > 0$, $Y > X > 0$, and the primes dividing $XY$ are precisely those in $R$. We consider $N$, the number of solutions of such an equation. Given a solution $(X, Y, z)$, let $D$ be the least positive integer such that $(XY/D)^{1/2}$ is an integer. Further, let $\omega$ be the number of distinct primes dividing $c$. Standard elementary approaches use an upper bound of $2^n$ for the number of possible $D$, and an upper bound of $2^{\omega-1}$ for the number of ideal factorizations of $c$ in the field $\mathbb{Q}(\sqrt{-D})$ which can correspond (in a standard designated way) to a solution in which $(XY/D)^{1/2} \in \mathbb{Z}$, and obtain $N \le 2^{n+\omega-1}$. Here we improve this by finding an inverse proportionality relationship between a bound on the number of $D$ which can occur in solutions and a bound (independent of $D$) on the number of ideal factorizations of $c$ which can correspond to solutions for a given $D$. We obtain $N \le 2^{n-1}+1$. The bound is precise for $n<4$: there are several cases with exactly $2^{n-1} + 1$ solutions.Comment: revised and expanded version of a paper published in Publ. Math. Debrecen, Vol. 93, issue 3-4, 2018, pp. 457-473 (2018

Smallest examples of strings of consecutive happy numbers

Abstract A happy number N is defined by the condition S n (N ) = 1 for some number n of iterations of the function S, where S(N ) is the sum of the squares of the digits of N . Up to 10 20 , the longest known string of consecutive happy numbers was length five. We find the smallest string of consecutive happy numbers of length 6, 7, 8, . . . , 13. For instance, the smallest string of six consecutive happy numbers begins with N = 7899999999999959999999996. We also find the smallest sequence of 3-consecutive cubic happy numbers of lengths 4, 5, 6, 7, 8, and 9