Giuga number

The Giuga numbers are natural numbers named after the mathematician Giuseppe Giuga with special properties. They are important in connection with a characterization of the prime numbers that he suspects . Related to the Giuga numbers are the primarily pseudo-perfect numbers and the Carmichael numbers .

Giuga's guess

In 1950 G. Giuga expressed the conjecture that a natural number is a prime number if and only if ${\ displaystyle n}$

{\ displaystyle \ sum _ {k = 1} ^ {n-1} k ^ {n-1} \ equiv -1 {\ pmod {n}}}

applies. For prime numbers this property follows from Fermat's little theorem . It is still unclear whether the reverse direction also applies. So it is not known whether there are also composite numbers with this property. According to a result from 1994, such a number should have more than 10,000 decimal places.

Giuga's conjecture is equivalent to the following statement: No natural number is both Giuga and Carmichael numbers .

It is also equivalent to (conjecture of Giuga and Takashi Ago ): n is prime if and only if

{\ displaystyle nB_ {n-1} \ equiv -1 {\ pmod {n}}.}

with the Bernoulli numbers . ${\ displaystyle B_ {n}}$

definition

A composite number is called a Giuga number if the following applies to all prime divisors of : divides . ${\ displaystyle n}$ ${\ displaystyle p}$ ${\ displaystyle n}$ ${\ displaystyle p}$ ${\ displaystyle {\ frac {n} {p}} - 1}$

The Carmichael numbers , which are related to the Giuga numbers, have a similar characterization: A composite number is called a Carmichael number if the following applies to all prime divisors of : divides . ${\ displaystyle p}$ ${\ displaystyle n}$ ${\ displaystyle \! \, p-1}$ ${\ displaystyle {\ frac {n} {p}} - 1}$

Equivalent characterizations

The Giuga numbers can be characterized in other ways: Let be a composite number and the set of prime divisors of . Then: ${\ displaystyle n}$ ${\ displaystyle P}$ ${\ displaystyle n}$

The number is exactly then giuga number if: . ${\ displaystyle n}$ ${\ displaystyle \ sum _ {k = 1} ^ {n-1} k ^ {\ varphi (n)} \ equiv -1 {\ pmod {n}}}$
The number is a Giuga number if and only if: is square-free and ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle \ sum _ {p \ in P} {\ frac {1} {p}} - \ prod _ {p \ in P} {\ frac {1} {p}} \ in \ mathbb {N}}$

This shows the close relationship of the Giuga numbers to the primarily pseudo-perfect numbers, which are characterized by.

{\ displaystyle \ sum _ {p \ in P} {\ frac {1} {p}} + \ prod _ {p \ in P} {\ frac {1} {p}} = 1}

The number is exactly then giuga number if: . ${\ displaystyle n}$ ${\ displaystyle nB _ {\ varphi (n)} \ equiv -1 {\ pmod {n}}}$

Here called the Euler's φ function and the Bernoulli numbers . ${\ displaystyle \ varphi}$ ${\ displaystyle B}$

Examples

Example 1:

Be ${\ displaystyle n: = 30.}$

Then the prime divisor has and . The following applies: ${\ displaystyle n}$ ${\ displaystyle p = 2.3}$ ${\ displaystyle 5}$

{\ displaystyle {\ begin {aligned} p = 2 & \ quad {\ textrm {divides}} & {\ frac {n} {p}} - 1 = {\ frac {30} {2}} - 1 & = & 15- 1 & = & 14 \\ p = 3 & \ quad {\ textrm {divides}} & {\ frac {n} {p}} - 1 = {\ frac {30} {3}} - 1 & = & 10-1 & = & 9 \ \ p = 5 & \ quad {\ textrm {divides}} & {\ frac {n} {p}} - 1 = {\ frac {30} {5}} - 1 & = & 6-1 & = & 5 \ end {aligned} }}

Thus is a Giuga number. ${\ displaystyle n = 30}$

Example 2:

The first seven Giuga numbers are:

30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838… (Follow A007850 in OEIS )

Well-known Giuga numbers

3 factors:
- 30 = 2 * 3 * 5
4 factors:
- 858 = 2 * 3 * 11 * 13
- 1722 = 2 * 3 * 7 * 41
5 factors:
- 66.198 = 2 * 3 * 11 * 17 * 59
6 factors:
- 2,214,408,306 = 2 * 3 * 11 * 23 * 31 * 47,057
- 24,423,128,562 = 2 * 3 * 7 * 43 * 3041 * 4447
7 factors:
- 432,749,205,173,838 = 2 * 3 * 7 * 59 * 163 * 1381 * 775,807
- 14,737,133,470,010,574 = 2 * 3 * 7 * 71 * 103 * 67,213 * 713,863
- 550.843.391.309.130.318 = 2 * 3 * 7 * 71 * 103 * 61.559 * 29.133.437
8 factors:
- 244,197,000,982,499,715,087,866,346 = 2 * 3 * 11 * 23 * 31 * 47,137 * 28,282,147 * 3,892,535,183
- 554.079.914.617.070.801.288.578.559.178 = 2 * 3 * 11 * 23 * 31 * 47.059 * 2.259.696.349 * 110.725.121.051
- 1,910,667,181,420,507,984,555,759,916,338,506 = 2 * 3 * 7 * 43 * 1831 * 138,683 * 2,861,051 * 1,456,230,512,169,437
10 factors:
- 4,200,017,949,707,747,062,038,711,509,670,656,632,404,195,753,751,630,609,228,764,416,142,557,211,582,098,432,545,190,323,474,818 = 2 * 3 * 11 * 23 * 31 * 47,059 * 2,217,342,227 * 1,729,101,023,519 * 8,491,659,218,261,819,498,490,029,296,021 * 58,254,480,569,119,734,123,541,298,976,556,403

properties

All Giuga numbers are free of squares.
All Giuga numbers are abundant .
There are only finitely many Giuga numbers with a given number of prime factors.
It is not known whether there are infinitely many Giuga numbers.
All known Giuga numbers are even. An odd Giuga number should consist of at least 14 prime factors. Since all Carmichael numbers are odd, Giuga's conjecture would also be proven if one could prove that all Giuga numbers are even.

literature

G. Giuga: Su una presumibile proprietà caratteristica dei numeri primi . Is. Lombardo Sci. Lett. Rend. A, 83 : 511-528, 1950
T. Agoh: On Giuga's conjecture . Manuscripta Math. 87 (4): 501-510, 1995
D. Borwein, JM Borwein, PB Borwein and R. Girgensohn: Giuga's Conjecture on Primality . Amer. Math. Monthly 103 : 40-50, 1996
Sorini L. "Un Metodo Euristico per la Soluzione della Congettura di Giuga", Facoltà di Economia, Università degli Studi di Urbino Carlo Bo, Quaderni di Economia, Matematica e Statistica, n. 68 , Ottobre (2001) ISSN 1720-9668.