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To begin with, i would like to apologize if my question is not up to the level of this forum. I have tried asking a variant of the following question on math.stackexchange.com and my question generated some comments (even one upvote) but no answers, so i decided to give it a shot over here.

My original question was:

Fermat's primality test for base 2 permits Poulet numbers to pass the test, as follows: ($2^x$−2)/$x$. Fermat's primality test in different bases will act as a sieve for eliminating most pseudo primes from passing the test, unless the numbers are Carmichael numbers.

I ran an experiment for the following formula ($5^x$−$3^x$−$2^x$)/$x$ and it seems to eliminate all but Carmichael numbers, without having to check different bases.I was capable of running the experiment until 10000 only (due to my lack of computing calculation power).Does anyone know about this formula and whether it still holds forever?

One of the comments mentioned that "25326001 is a (strong) pseudoprime for the bases 2,3,5, thus it will pass your test. But it is not a Carmichael number."

I have then asked if that will be the smallest number that is not a carmicahel number to pass the test?

And i received this following comment: "If you check larger numbers, more pseudoprimes that aren't Carmichael numbers should turn up alongside the Carmichael numbers. But that takes more computational power"

So my question is whether anyone knows if 25326001 is the first non carmichael number to pass the test or no?

Again, my apologies if i am interrupting the level of this forum, but my goal is simply to get an answer.

Thanks,

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    $\begingroup$ Link to the question on math.stackexchange, please? And please leave a link there to the question here. $\endgroup$
    – Gerry Myerson
    Commented Aug 18, 2020 at 5:39
  • $\begingroup$ $25326001$ is the smallest number that's a strong pseudoprime to all three bases $2$, $3$, and $5$ (which doesn't quite answer your question). $\endgroup$
    – Gerry Myerson
    Commented Aug 18, 2020 at 5:45
  • $\begingroup$ I will do the same there, here is the link: math.stackexchange.com/questions/3789932/… $\endgroup$
    – Ilan Alon
    Commented Aug 18, 2020 at 5:45

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The answer is No. E.g., see OEIS A153580. for smaller examples.

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