Higgs prime

A Higgs prime, named after Denis Higgs, is a prime number with a totient (one less than the prime) that evenly divides the square of the product of the smaller Higgs primes. (This can be generalized to cubes, fourth powers, etc.) To put it algebraically, given an exponent a, a Higgs prime Hpn satisfies

\( \phi(Hp_n)|\prod_{i = 1}^{n - 1} {Hp_i}^a\mbox{ and }Hp_n > Hp_{n - 1} \)

where Φ(x) is Euler's totient function.

For squares, the first few Higgs primes are 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, ... (sequence A007459 in the 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:

Exponent	75th Higgs prime	Not Higgs prime below 75th Higgs prime
2	797	17, 41, 73, 83, 89, 97, 103, 109, 113, 137, 163, 167, 179, 193, 227, 233, 239, 241, 251, 257, 271, 281, 293, 307, 313, 337, 353, 359, 379, 389, 401, 409, 433, 439, 443, 449, 457, 467, 479, 487, 499, 503, 521, 541, 563, 569, 577, 587, 593, 601, 613, 617, 619, 641, 647, 653, 673, 719, 739, 751, 757, 761, 769, 773
3	509	17, 97, 103, 113, 137, 163, 193, 227, 239, 241, 257, 307, 337, 353, 389, 401, 409, 433, 443, 449, 479, 487
4	409	97, 193, 257, 353, 389
5	389	193, 257
6	383	257
7	383	257

Observation further reveals that a Fermat prime \( 2^{{2^{n}}}+1 \) can't be a Higgs prime for the ath power if a is less than 2n.

It's not known if there are infinitely many Higgs primes for any exponent a greater than 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). Burris & Lee (1993) 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

Burris, S.; Lee, S. (1993). "Tarski's high school identities". Amer. Math. Monthly. 100 (3): 231–236 [p. 233]. JSTOR 2324454.
Sloane, N.; Plouffe, S. (1995). The Encyclopedia of Integer Sequences. New York: Academic Press. ISBN 0-12-558630-2. M0660

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Prime number classes
By formula

Fermat (22n + 1) Mersenne (2p − 1) Double Mersenne (22p−1 − 1) Wagstaff (2p + 1)/3 Proth (k·2n + 1) Factorial (n! ± 1) Primorial (pn# ± 1) Euclid (pn# + 1) Pythagorean (4n + 1) Pierpont (2m·3n + 1) Quartan (x4 + y4) Solinas (2m ± 2n ± 1) Cullen (n·2n + 1) Woodall (n·2n − 1) Cuban (x3 − y3)/(x − y) Carol (2n − 1)2 − 2 Kynea (2n + 1)2 − 2 Leyland (xy + yx) Thabit (3·2n − 1) Williams ((b−1)·bn − 1) Mills (⌊A3n⌋)

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Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent

Happy Dihedral Palindromic Emirp Repunit (10n − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Tetradic

Patterns

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By size

Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known

Complex numbers

Eisenstein prime Gaussian prime

Composite numbers

Pseudoprime
Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious