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 $m=n$.



Some equations similar to Goormaghtigh problem
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}$


