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I read an article that describes how to distinguish between real and fake sequences of coin tosses, with good reliability: we should check the longest run of heads (in real sequences of length $n$, the longest run of heads is about $\log_2 n-1$, so if a sequence's longest run of heads is much different than that, there is reason to suspect that the sequence is fake). If that's not enough, we could use more sophisticated checks, such as counting the successive ordered pairs of results.

That made me wonder about how to distinguish between real and fake prime gap sequences.

Suppose an alien from another planet gives you a sequence, claiming that it is the $1000$ consecutive prime gaps starting from the $10^{100}$th prime gap. How could we check if the alien is likely to be lying?

(With current human technology, we are unable to find the $10^{100}$th prime number.)

I know of the following checks:

What are some other checks?

Remarks:

Of course, there is no way to be sure that any given sequence of coin tosses is fake, and there is usually no way to be sure that the alien is lying (so I have included the "soft-question" tag). But in both cases, we can make good guesses based on known mathematics.

I picked the numbers $1000$ and $10^{100}$, but there is nothing particularly special about them; feel free to adjust them, if that would allow you to provide a more insightful answer.

One might assume that if the alien were lying, it could still anticipate how we would check, and thus avoid detection; but let's not assume that the alien is that sophisticated.

Dan
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    Well, take any given prime $q$. Then your (unknown) starting prime $p$ has some non-zero residue $\overline p \pmod q$. Then ${\overline p +g_i}\pmod q$ can be computed (for each of the possible $\overline p$) and can never be $0$. You can test this for the first couple hundred primes $q$. I'd say it's hard to avoid hitting $0$ accidentally if you aren't making a real effort. – lulu Jun 19 '23 at 09:00
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    Of course, one can avoid it with effort. Instead of the prime gap sequence, the alien might have taken the gaps between numbers prime to the first $10^{10}$ primes, or whatever. That sequence will pass my test, clearly. More broadly, the alien can take the gap sequence between very convincing pseudo primes. – lulu Jun 19 '23 at 09:04
  • Using @lulu's suggestion, with almost any small prime $q$ you'd find that all but one residue is used mod $q$, from which you can deduce $p \bmod q$ where $p$ is the alleged $10^{100}$th prime. With enough such small primes $q$, you can calculate $p$ using the Chinese remainder theorem. You could then use primality tests to check the numbers starting from $p$. You still wouldn't know if $p$ really was the $10^{100}$th prime, though you can predict roughly what size it should be. – Jaap Scherphuis Jun 19 '23 at 13:22
  • @JaapScherphuis $p$ would be immense, over $100$ digits. Can we actually calculate it, practically speaking? Using primality tests to find the next $999$ primes also seems like a very tall order. – Dan Jun 19 '23 at 13:54
  • @JaapScherphuis OK, I just did some tests on Wolfram, and now I realize that Wolfram can tell us if a given $10^7$-digit number is prime or not. But as I mentioned in my question, we can adjust the given number $10^{100}$: what if it were large enough so that we couldn't use primality tests to check the numbers starting from $p$ ? – Dan Jun 19 '23 at 14:18
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    Wolfram can check a $10^7$-digit number for primality only in exceptional cases , for example when it has a small prime factor. – Peter Jun 19 '23 at 17:38
  • It might be possible to find the $10^{100}$ th prime at some time in the future. It is however far far beyond the current record. – Peter Jun 19 '23 at 17:39
  • @Peter But the alien only gives us (supposed) prime gaps, not primes themselves. – Dan Jul 03 '23 at 14:41
  • OK , now I have understood how the question is meant. – Peter Jul 03 '23 at 14:51

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