### One of a continuing series

The following number has been demonstrated to be prime:

3756801695685·2^{666669}+1

That is 3756801695685 times (2 raised to the 666669 power). Plus 1. Here’s what you need to do.

Write this number out as one long string of digits. Now create another number by scrambling the digits of the first number. One number will be larger than the other. Subtract the smaller from the larger to produce another number. Call this number P.

Prove that P is not prime. It’s easy. Post your answer as a comment below.

## Update and solution

It’s time. A few people have commented, and Prasad has made a stab at a solution. He was on the right track, but partial credit is not awarded. Here’s the solution, but first you have to read through an old joke:

Question: What has four legs, barks, and is full of concrete.

Answer: It’s a dog. I put the concrete in to make it hard.

And that’s what I did with this week’s Quiz Question. I put in something to make an easy problem hard, and that something is all this nonsense about large prime numbers. Forget about it. It doesn’t make any difference. The following will work with any decimal integer.

- Make up a number: 123456.
- Scramble the digits: 346521.
- Subtract the smaller from the larger: 345621 – 123456 = 222165.
- 222165 is not prime. It’s divisible by 9.

Prasad did not post a comment, but on Facebook he noted that if you “flip the first digit.” Actually he should have said “transpose the first two digits.” That will produce a number ending in 0, which will be divisible by 10, of course. It will also produce a number divisible by 9. However, the problem requires the difference to be non-prime for all permutations of the digits, and that is the case.

I will provide a proof upon request.

Ok, so 2^222= 6739986666787659948666753771754907668409286105635143120275902562304

And you want me to write out 2^666,669 then multiply by 3756801695685…. and claim it’s easy? I don’t think you can write this number on a roll of toilet paper….

🙂

I cannot possibly execute step one… how exactly is this “Easy?”

Stirling,

Thanks for commenting. You are the first to notice the impossibility of following my instructions literally. Now that you have broken the ice, here is a hint. Do not. I repeat, do not attempt to write out this number on a piece of paper. However, just imagine that you are doing it. This is, after all, mathematics and not engineering.

Another hint: Pretend you could/did write out the first number. Pretend that you could/did generate the second number by scrambling the digits of the first number. Pretend that you could/did compute the difference to get the number called P. Now answer the question. Prove that P is not prime.

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