Is the occurence of $4653$ in both expressions a coincidence?

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The smallest composite strong-probable prime to base $2$ greater than $10^5$ is $$10^5+4653$$ and the smallest composite strong-probable prime to base $2$ greater than $10^6$ is $$10^6+4653$$

Is it a coincidence that we have the number $4653$ in both expressions or is there an explanation ?

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The sequence of strong pseudoprimes to base $2$ is given by A001262,

$$2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281\dots$$

What we are looking for are two entries of form,

$$a^m+b\\a^n+b$$

Using just the first $10000$ entries, we are bound to find such pairs. Hence, we have your,

$$10^5+4653,\quad 10^6+4653$$

as well as,

$$10^3 + \color{blue}{163}\times6027,\quad 10^5 + \color{blue}{163}\times6027$$

$$9^2 + 3952,\quad 9^3 + 3952$$

$$8^5 + 45067409,\quad 8^9 + 45067409$$

$$7^4 + 1323442, \quad 7^6 + 1323442$$

and so on. (Note the surprising cameo played by my favorite number $163$.)

In conclusion, these pairs are probably just coincidences, though the first one is noticeable because of our base-$10$ notation.