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# Higgs prime

A Higgs prime is a prime number $Hp_{n}$ for which, given an exponent $a$, it is the case that

$\phi(Hp_{n})|\prod_{{i=1}}^{{n-1}}{Hp_{i}}^{a},$ |

(where $\phi(x)$ is Euler’s totient function) and $Hp_{n}>Hp_{{n-1}}$.

For $a=2$, the first few Higgs primes are 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, etc., listed in A007459 of Sloane’s OEIS. So, for example, 13 is a Higgs prime because the square of the product of the smaller Higgs primes is 5336100, and divided by 12 this is 444675. But 17 is not a Higgs prime because the square of the product of the smaller primes is 901800900, which leaves a remainder of 4 when divided by 16.

From observation of the first few Higgs primes for squares through seventh powers, it would seem more compact to list those primes that are not Higgs primes. Observation further reveals that a Fermat prime $2^{{2^{n}}}+1$ can’t be a Higgs prime for the $a$th power if $a<2^{n}$.

It’s not known if there are infinitely many Higgs primes for any exponent $a>1$. The situation is quite different for $a=1$. There are only four of them: 2, 3, 7 and 43 (a sequence suspiciously similar to Sylvester’s sequence). In 1993, Burris and Lee found that about a fifth of the primes below a million are Higgs prime, and they concluded that even if the sequence of Higgs primes for squares is finite, “a computer enumeration is not feasible.”

# References

- 1 S. Burris & S. Lee, “Tarski’s high school identities”, Amer. Math. Monthly 100 (1993): 233
- 2 N. Sloane & S. Plouffe, The Encyclopedia of Integer Sequences, New York: Academic Press (1995): M0660

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11A41*no label found*

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## Comments

## Clarifications

Wouldn't it be simpler to replace \phi(Hp_n) with Hp_n - 1 and the condition \pi(Hp_n) > \pi(Hp_{n-1}) with Hp_n > Hp_{n-1}?

Since Hp_n is prime, it seems that these are equivalent and simpler formulations.

Also, it would be nice to have some motivation for the definition -- do these sequences of primes arise from some other considerations?

## Re: Clarifications about Higgs primes

Per your suggestion I changed \pi(Hp_n) > \pi(Hp_{n-1}) to Hp_n > Hp_{n-1}. I don't blame anyone for thinking that was a symbological obfuscation, there really was no good reason for it. Thank you for pointing this out.

As for \phi(Hp_n) to Hp_n - 1, I remember thinking yesterday that this wasn't an obfuscation and that there was a very good reason to put it this way to bring out some relation to cototient valences for which some overnight calculations are required. Unfortunately my computer crashed some time around midnight in the midst of the calculation and today I can't remember for the life of me what that was, or if maybe it was in connection to some other totient topic. I've managed to obfuscate this to myself! Perhaps one of the younger PM users could refresh my memory if this has something to do with something I've chatted with them about.

For me it's become quite second-nature that \phi(p) = p - 1, but perhaps I should mention it here just the same (I mentioned it at Wikipedia, where it is less likely a casual reader would figure it out on first sight).

In regards to your last point, I became interested in these primes because of the ways in which they are not Fermat primes. Burris and Lee studied these in connection to "high-school algebras," something which I personally couldn't care less about, as I passed high school algebra decades ago.

## Re: Clarifications about Higgs primes

> I can't remember for the life of me what that was, or if maybe it was in connection to some other totient topic. I've managed to obfuscate this to myself! Perhaps one of the younger PM users could refresh my memory if this has something to do with something I've chatted with them about.

>

You and I have talked about the highly cototient primes, and besides their relation to primorials, we haven't been able to uncover anything else about them. You've never talked to me about Higgs primes, and since there are so few of them for 1 and so many of them for 2, I doubt they can provide any insight into this. I hope that somehow helps you remember the totient application of Higgs primes that you've forgotten.