Higgs prime number

In number theory , a Higgs prime number for the power a is a prime number where the -th power divides the product of all smaller Higgs prime numbers. In algebraic terms, given a power , this means that the Higgs prime number fulfills the following condition: ${\displaystyle Hp_{n}\in \mathbb {P} }$ ${\displaystyle Hp_{n}-1}$ ${\displaystyle a}$ ${\displaystyle a\in \mathbb {N} }$ ${\displaystyle Hp_{n}}$

${\displaystyle \varphi (Hp_{n})=Hp_{n}-1{\mbox{ teilt }}\prod _{i=1}^{n-1}{Hp_{i}}^{a}{\mbox{ mit }}Hp_{n}>Hp_{n-1}}$

where is Euler's Phi function (for each natural number it indicates how many coprime natural numbers there are that are not greater than ; for prime numbers is ). ${\displaystyle \varphi (n)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle \varphi (p)=p-1}$

The Higgs prime numbers were named after the British mathematician Denis Higgs .

Examples

• The first Higgs prime numbers for the power (i.e. for squares) are the following: ${\displaystyle a=2}$

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349, 367, 373, 383, 397, 419, 421, 431, 461, 463, 491, 509, 523, 547, 557, 571, ... (sequence A007459 in OEIS )

• The number is a Higgs prime number for the power , because the square of the product of the smaller Higgs prime numbers has (it is ) the number as a divisor . ${\displaystyle Hp_{8}=23}$ ${\displaystyle a=2}$ ${\displaystyle (2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 19)^{2}=325550124900}$ ${\displaystyle Hp_{8}-1=23-1=22}$ ${\displaystyle 325550124900:22=14797732950}$
• The number is not a Higgs prime to the power : the square of the product of the smaller Higgs prime numbers, so the number doesn't have a divisor (the remainder remains ). ${\displaystyle 17}$ ${\displaystyle a=2}$ ${\displaystyle (2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13)^{2}=901800900}$ ${\displaystyle 17-1=16}$ ${\displaystyle 4}$
• The number is a Higgs prime number for the power , because the -th power of the product of the smaller Higgs prime numbers, i.e. the number as a divisor (it is ). ${\displaystyle 13}$ ${\displaystyle a=6}$ ${\displaystyle 6}$ ${\displaystyle (2\cdot 3\cdot 5\cdot 7\cdot 11)^{6}=151939915084881000000}$ ${\displaystyle 13-1=12}$ ${\displaystyle 151939915084881000000:12=12661659590406750000}$
• With higher powers , more and more prime numbers are also Higgs prime numbers at the same time, so that it makes sense to specify those prime numbers that are not Higgs prime numbers at the same time. The following table shows these "non-Higgs prime numbers" for a given power up to the 100th Higgs prime number for the respective power : ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle a}$

exponent ${\displaystyle a}$	100. Higgs prime number	no Higgs prime numbers for the power up to the 100th Higgs prime number of this power ${\displaystyle a}$
2	1117	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, 809, 811, 821, 823, 857, 877, 881, 887, 919, 929, 937, 953, 971, 977, 997, 1009, 1021, 1031, 1033, 1049, 1069, 1091, 1097 (a total of 87 prime numbers)
3	733	17, 97, 103, 113, 137, 163, 193, 227, 239, 241, 257, 307, 337, 353, 389, 401, 409, 433, 443, 449, 479, 487, 577, 593, 613, 619, 641, 647, 653, 673 (total of 30 prime numbers)
4th	593	97, 193, 257, 353, 389, 449, 487, 577 (8 prime numbers in total)
5	563	193, 257, 449
6th	547	257
7th	547	257
8th	541	---

properties

• There are only four Higgs prime numbers for the power : ${\displaystyle a=1}$

2, 3, 7, 43

Proof:

Suppose there is a prime number (which is closest ) which is a Higgs prime to the power . Then there must be a divisor of all previous Higgs prime numbers for the power , i.e. of . But this cannot be the case because there cannot be a divisor of the smaller number . Thus all prime numbers are eliminated. All prime numbers are eliminated by simple computer calculations. ${\displaystyle p>2\cdot 3\cdot 7\cdot 43=1806}$ ${\displaystyle p=1811}$ ${\displaystyle a=1}$ ${\displaystyle p-1\geq 1810}$ ${\displaystyle a=1}$ ${\displaystyle (2\cdot 3\cdot 7\cdot 43)^{1}=1806}$ ${\displaystyle p-1\geq 1810}$ ${\displaystyle 1806}$ ${\displaystyle p\geq 1811}$ ${\displaystyle 1811>p>43}$ ${\displaystyle \Box }$

• All known Fermat prime numbers are not Higgs prime numbers for the -th powers with . ${\displaystyle 2^{2^{n}}+1}$ ${\displaystyle a}$ ${\displaystyle a<2^{n}}$

Proof:

You can calculate that relatively quickly using a computer

• the first Fermat prime number is not a Higgs prime for . ${\displaystyle 2^{2^{0}}+1=2^{1}+1=3}$ ${\displaystyle a<2^{0}=1}$
• the second Fermat prime number is not a Higgs prime for . ${\displaystyle 2^{2^{1}}+1=2^{2}+1=5}$ ${\displaystyle a<2^{1}=2}$
• the third Fermat prime number is not a Higgs prime for . ${\displaystyle 2^{2^{2}}+1=2^{4}+1=17}$ ${\displaystyle a<2^{2}=4}$
• the fourth Fermat prime number is not a Higgs prime for . ${\displaystyle 2^{2^{3}}+1=2^{8}+1=257}$ ${\displaystyle a<2^{3}=8}$
• the fifth and last known Fermat prime number is not a Higgs prime for . ${\displaystyle 2^{2^{4}}+1=2^{1}6+1=65537}$ ${\displaystyle a<2^{4}=16}$ ${\displaystyle \Box }$

• About a fifth of the prime numbers out of a million are Higgs prime numbers.

The discoverers of this property concluded that, even if the number of Higgs prime numbers for the power is finite, “computer counting is not possible”. ${\displaystyle a=2}$

Unsolved problems

• It is not known whether there are infinitely many Higgs prime numbers for exponents . ${\displaystyle a\geq 2}$

Individual evidence

1. ^ Stanley Burris, Simon Lee: Tarski's high school identities. American Mathematical Monthly 100 (3), 1993, pp. 231-236 , accessed July 2, 2018 .