# Some equations similar to Goormaghtigh problem

Let $m,n,t$ be some primes and $x$ and $y$ be some prime powers. Also suppose that $mn\mid(t-1)$. Which one of the following is correct?

1- If $t=\dfrac{x^{n}-1}{(x-1)(x-1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then $x=y$ and $m=n$.

2- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then there is no solution for $x,y,m,n$.

3- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}+1}{(y+1)(y+1,m)}$, then $x=y$ and $m=n$.

There are counter-examples to all three claims in the question above, so none of the claims are correct.

1- A counter example is {x,n,y,m,t}={2,5,5,3,31} as was indicated in a comment above.

Another counter example is {x,n,y,m,t}={2,3,13,2,7}.

2- A counter example is {x,n,y,m,t}={3,3,13,2,7}.

Another counter example is {x,n,y,m,t}={5,3,13,2,7}.

3- A counter example is {x,n,y,m,t}={2,7,7,3,43}.

For those that enjoy searching for patterns in data, the table below for $(x^n,y^m)\le 10,000$ provides a bit more to contemplate. The table includes cases that violate the assumption $m\,n|(t-1)$ and trivial cases of Claims 1 and 2. The non-trivial cases that satisfy the assumption $m\,n|(t-1)$ are contained in rows 38 to 56 below. I've currently generated results for $(x^n,y^m)\le 100,000,000$ which is too much data to post here.

$\begin{array}{ccccccccccccccccc} \# & \text{mn$|$(t-1)} & \text{Trivial} & \text{Claim} & \text{x=y} & \text{n=m} & \text{x} & \text{n} & \text{y} & \text{m} & \text{t} & \text{$x^n$} & \text{$y^m$} & \{\text{Factor(x)}\} & \{\text{Factor(y)}\} & \{\text{Factor($x^n\pm$1)}\} & \{\text{Factor($y^m\pm$1 )}\} \\ 1 & \text{False} & \text{False} & 1 & \text{False} & \text{True} & 2 & 2 & 5 & 2 & 3 & 4 & 25 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ \end{array} \right) \\ 2 & \text{False} & \text{False} & 1 & \text{False} & \text{True} & 2 & 3 & 4 & 3 & 7 & 8 & 64 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 7 & 1 \\ \end{array} \right) \\ 3 & \text{False} & \text{False} & 2 & \text{False} & \text{True} & 3 & 3 & 2 & 3 & 7 & 27 & 8 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) \\ 4 & \text{False} & \text{False} & 2 & \text{False} & \text{True} & 3 & 3 & 4 & 3 & 7 & 27 & 64 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 7 & 1 \\ \end{array} \right) \\ 5 & \text{False} & \text{False} & 2 & \text{False} & \text{True} & 4 & 3 & 3 & 3 & 13 & 64 & 27 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 13 & 1 \\ \end{array} \right) \\ 6 & \text{False} & \text{False} & 2 & \text{False} & \text{True} & 5 & 3 & 2 & 3 & 7 & 125 & 8 & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) \\ 7 & \text{False} & \text{False} & 2 & \text{False} & \text{True} & 5 & 3 & 4 & 3 & 7 & 125 & 64 & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 7 & 1 \\ \end{array} \right) \\ 8 & \text{False} & \text{False} & 3 & \text{False} & \text{True} & 3 & 3 & 5 & 3 & 7 & 27 & 125 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 7 & 1 \\ \end{array} \right) \\ 9 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 2 & 2 & 2 & 2 & 3 & 4 & 4 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) \\ 10 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 2 & 3 & 2 & 3 & 7 & 8 & 8 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) \\ 11 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 2 & 5 & 2 & 5 & 31 & 32 & 32 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 31 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 31 & 1 \\ \end{array} \right) \\ 12 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 2 & 7 & 2 & 7 & 127 & 128 & 128 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 127 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 127 & 1 \\ \end{array} \right) \\ 13 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 2 & 13 & 2 & 13 & 8191 & 8192 & 8192 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 8191 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 8191 & 1 \\ \end{array} \right) \\ 14 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 3 & 2 & 3 & 2 & 2 & 9 & 9 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) \\ 15 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 3 & 3 & 3 & 3 & 13 & 27 & 27 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 13 & 1 \\ \end{array} \right) \\ 16 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 3 & 7 & 3 & 7 & 1093 & 2187 & 2187 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 1093 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 1093 & 1 \\ \end{array} \right) \\ 17 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 4 & 3 & 4 & 3 & 7 & 64 & 64 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 7 & 1 \\ \end{array} \right) \\ 18 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 5 & 2 & 5 & 2 & 3 & 25 & 25 & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ \end{array} \right) \\ 19 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 5 & 3 & 5 & 3 & 31 & 125 & 125 & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 31 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 31 & 1 \\ \end{array} \right) \\ 20 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 13 & 2 & 13 & 2 & 7 & 169 & 169 & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 7 & 1 \\ \end{array} \right) \\ 21 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 13 & 3 & 13 & 3 & 61 & 2197 & 2197 & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 3 & 2 \\ 61 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 3 & 2 \\ 61 & 1 \\ \end{array} \right) \\ 22 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 37 & 2 & 37 & 2 & 19 & 1369 & 1369 & \left( \begin{array}{cc} 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \\ 19 & 1 \\ \end{array} \right) \\ 23 & \text{False} & \text{True} & 1 & \text{True} & \text{True} & 61 & 2 & 61 & 2 & 31 & 3721 & 3721 & \left( \begin{array}{cc} 61 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 61 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 5 & 1 \\ 31 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 5 & 1 \\ 31 & 1 \\ \end{array} \right) \\ 24 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 2 & 5 & 2 & 5 & 11 & 32 & 32 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 11 & 1 \\ \end{array} \right) \\ 25 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 2 & 7 & 2 & 7 & 43 & 128 & 128 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 43 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 43 & 1 \\ \end{array} \right) \\ 26 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 2 & 11 & 2 & 11 & 683 & 2048 & 2048 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 683 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 683 & 1 \\ \end{array} \right) \\ 27 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 2 & 13 & 2 & 13 & 2731 & 8192 & 8192 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 2731 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 2731 & 1 \\ \end{array} \right) \\ 28 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 3 & 3 & 3 & 3 & 7 & 27 & 27 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 7 & 1 \\ \end{array} \right) \\ 29 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 3 & 5 & 3 & 5 & 61 & 243 & 243 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 61 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 61 & 1 \\ \end{array} \right) \\ 30 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 3 & 7 & 3 & 7 & 547 & 2187 & 2187 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 547 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 547 & 1 \\ \end{array} \right) \\ 31 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 4 & 3 & 4 & 3 & 13 & 64 & 64 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ 13 & 1 \\ \end{array} \right) \\ 32 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 4 & 5 & 4 & 5 & 41 & 1024 & 1024 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 2 \\ 41 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 2 \\ 41 & 1 \\ \end{array} \right) \\ 33 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 5 & 3 & 5 & 3 & 7 & 125 & 125 & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 7 & 1 \\ \end{array} \right) \\ 34 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 5 & 5 & 5 & 5 & 521 & 3125 & 3125 & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 1 \\ 521 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 1 \\ 521 & 1 \\ \end{array} \right) \\ 35 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 7 & 3 & 7 & 3 & 43 & 343 & 343 & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 43 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 43 & 1 \\ \end{array} \right) \\ 36 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 13 & 3 & 13 & 3 & 157 & 2197 & 2197 & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 7 & 1 \\ 157 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 7 & 1 \\ 157 & 1 \\ \end{array} \right) \\ 37 & \text{False} & \text{True} & 3 & \text{True} & \text{True} & 16 & 3 & 16 & 3 & 241 & 4096 & 4096 & \left( \begin{array}{cc} 2 & 4 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ \end{array} \right) & \left( \begin{array}{cc} 17 & 1 \\ 241 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 17 & 1 \\ 241 & 1 \\ \end{array} \right) \\ 38 & \text{True} & \text{False} & 1 & \text{False} & \text{False} & 2 & 3 & 13 & 2 & 7 & 8 & 169 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 7 & 1 \\ \end{array} \right) \\ 39 & \text{True} & \text{False} & 1 & \text{False} & \text{False} & 2 & 5 & 5 & 3 & 31 & 32 & 125 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 31 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 31 & 1 \\ \end{array} \right) \\ 40 & \text{True} & \text{False} & 1 & \text{False} & \text{False} & 2 & 5 & 61 & 2 & 31 & 32 & 3721 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 61 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 31 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 5 & 1 \\ 31 & 1 \\ \end{array} \right) \\ 41 & \text{True} & \text{False} & 1 & \text{False} & \text{False} & 2 & 7 & 19 & 3 & 127 & 128 & 6859 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 127 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 3 \\ 127 & 1 \\ \end{array} \right) \\ 42 & \text{True} & \text{False} & 1 & \text{False} & \text{False} & 3 & 3 & 25 & 2 & 13 & 27 & 625 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 3 & 1 \\ 13 & 1 \\ \end{array} \right) \\ 43 & \text{True} & \text{False} & 1 & \text{False} & \text{False} & 4 & 3 & 13 & 2 & 7 & 64 & 169 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 7 & 1 \\ \end{array} \right) \\ 44 & \text{True} & \text{False} & 1 & \text{False} & \text{False} & 5 & 3 & 61 & 2 & 31 & 125 & 3721 & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 61 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 31 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 5 & 1 \\ 31 & 1 \\ \end{array} \right) \\ 45 & \text{True} & \text{False} & 1 & \text{False} & \text{False} & 7 & 3 & 37 & 2 & 19 & 343 & 1369 & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \\ 19 & 1 \\ \end{array} \right) \\ 46 & \text{True} & \text{False} & 1 & \text{False} & \text{True} & 4 & 2 & 9 & 2 & 5 & 16 & 81 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 5 & 1 \\ \end{array} \right) \\ 47 & \text{True} & \text{False} & 2 & \text{False} & \text{False} & 3 & 3 & 13 & 2 & 7 & 27 & 169 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 7 & 1 \\ \end{array} \right) \\ 48 & \text{True} & \text{False} & 2 & \text{False} & \text{False} & 3 & 5 & 13 & 3 & 61 & 243 & 2197 & \left( \begin{array}{cc} 3 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 61 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 3 & 2 \\ 61 & 1 \\ \end{array} \right) \\ 49 & \text{True} & \text{False} & 2 & \text{False} & \text{False} & 4 & 3 & 25 & 2 & 13 & 64 & 625 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 3 & 1 \\ 13 & 1 \\ \end{array} \right) \\ 50 & \text{True} & \text{False} & 2 & \text{False} & \text{False} & 4 & 5 & 81 & 2 & 41 & 1024 & 6561 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 2 \\ 41 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 5 \\ 5 & 1 \\ 41 & 1 \\ \end{array} \right) \\ 51 & \text{True} & \text{False} & 2 & \text{False} & \text{False} & 5 & 3 & 13 & 2 & 7 & 125 & 169 & \left( \begin{array}{cc} 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 1 \\ 7 & 1 \\ \end{array} \right) \\ 52 & \text{True} & \text{False} & 2 & \text{False} & \text{False} & 8 & 3 & 37 & 2 & 19 & 512 & 1369 & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 3 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \\ 19 & 1 \\ \end{array} \right) \\ 53 & \text{True} & \text{False} & 2 & \text{False} & \text{False} & 11 & 3 & 73 & 2 & 37 & 1331 & 5329 & \left( \begin{array}{cc} 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 73 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 3 & 2 \\ 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 3 & 2 \\ 37 & 1 \\ \end{array} \right) \\ 54 & \text{True} & \text{False} & 2 & \text{False} & \text{True} & 8 & 3 & 7 & 3 & 19 & 512 & 343 & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 3 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 19 & 1 \\ \end{array} \right) \\ 55 & \text{True} & \text{False} & 2 & \text{False} & \text{True} & 9 & 3 & 8 & 3 & 73 & 729 & 512 & \left( \begin{array}{cc} 3 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 5 & 1 \\ 73 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ 73 & 1 \\ \end{array} \right) \\ 56 & \text{True} & \text{False} & 3 & \text{False} & \text{False} & 2 & 7 & 7 & 3 & 43 & 128 & 343 & \left( \begin{array}{cc} 2 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 43 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ 43 & 1 \\ \end{array} \right) \\ 57 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 4 & 2 & 4 & 2 & 5 & 16 & 16 & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ \end{array} \right) \\ 58 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 7 & 3 & 7 & 3 & 19 & 343 & 343 & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 2 \\ 19 & 1 \\ \end{array} \right) \\ 59 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 8 & 3 & 8 & 3 & 73 & 512 & 512 & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ 73 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 7 & 1 \\ 73 & 1 \\ \end{array} \right) \\ 60 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 9 & 2 & 9 & 2 & 5 & 81 & 81 & \left( \begin{array}{cc} 3 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 5 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 5 & 1 \\ \end{array} \right) \\ 61 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 16 & 2 & 16 & 2 & 17 & 256 & 256 & \left( \begin{array}{cc} 2 & 4 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ 17 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ 17 & 1 \\ \end{array} \right) \\ 62 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 17 & 3 & 17 & 3 & 307 & 4913 & 4913 & \left( \begin{array}{cc} 17 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 17 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 307 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 307 & 1 \\ \end{array} \right) \\ 63 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 19 & 3 & 19 & 3 & 127 & 6859 & 6859 & \left( \begin{array}{cc} 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 3 \\ 127 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 3 & 3 \\ 127 & 1 \\ \end{array} \right) \\ 64 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 25 & 2 & 25 & 2 & 13 & 625 & 625 & \left( \begin{array}{cc} 5 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 3 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 3 & 1 \\ 13 & 1 \\ \end{array} \right) \\ 65 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 73 & 2 & 73 & 2 & 37 & 5329 & 5329 & \left( \begin{array}{cc} 73 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 73 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 3 & 2 \\ 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 4 \\ 3 & 2 \\ 37 & 1 \\ \end{array} \right) \\ 66 & \text{True} & \text{True} & 1 & \text{True} & \text{True} & 81 & 2 & 81 & 2 & 41 & 6561 & 6561 & \left( \begin{array}{cc} 3 & 4 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 5 \\ 5 & 1 \\ 41 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 5 \\ 5 & 1 \\ 41 & 1 \\ \end{array} \right) \\ 67 & \text{True} & \text{True} & 3 & \text{True} & \text{True} & 8 & 3 & 8 & 3 & 19 & 512 & 512 & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 3 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 3 \\ 19 & 1 \\ \end{array} \right) \\ 68 & \text{True} & \text{True} & 3 & \text{True} & \text{True} & 9 & 3 & 9 & 3 & 73 & 729 & 729 & \left( \begin{array}{cc} 3 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 5 & 1 \\ 73 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 1 \\ 5 & 1 \\ 73 & 1 \\ \end{array} \right) \\ 69 & \text{True} & \text{True} & 3 & \text{True} & \text{True} & 11 & 3 & 11 & 3 & 37 & 1331 & 1331 & \left( \begin{array}{cc} 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 3 & 2 \\ 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 2 & 2 \\ 3 & 2 \\ 37 & 1 \\ \end{array} \right) \\ \end{array}$