Primes or not?

I’m in the process of clearing out old bookmarks, and stumbled on this puzzle from @jase_jwanner:

I shall give you a moment to ponder these, and put my spoiler below the line.


The first and last

The first and last of these are, if you look at them the right way, completely obvious: the two numbers in $23567897614^2 - 1$ are both squares, and what you have is a difference thereof, so this is clearly a composite number. (In fact, a moment’s thought tells you it’s a multiple of 5.)

Similarly, $345643554^{10} - 169$ is a difference of two squares, so it’s composite.

The second

Less obviously, but still similarly, $34564344^3 -1$ is a difference of two cubes, and $x^3 - 1 = (x-1)(x^2 + x + 1)$ - so this is also composite.

The third

Now, $76543556556625731$ is the interesting one. From the context of the tweet, I would imagine it’s going to be composite, because there’s no simple way to show that it’s prime.

There are no obvious squares nearby, so my next trick from the bag is to look at small factors.

And we’re in luck! Summing the digits gives 81, which - being a multiple of 9 - tells us that $76543556556625731$ is a multiple of 9.

Did you tackle them any othe ways? Let me know in the comments!

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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