Giuga number

The Giuga numbers are named after the mathematician Giuseppe Giuga natural numbers with special properties. They are associated with a suspected of it characterization of prime importance. Relates to the Giuga numbers are the primary pseudo- perfect numbers and the Carmichael numbers.

Giugas presumption

In 1950, G. Giuga expressed the belief that a natural number is a prime number if and only if the following holds. For primes follows this property from the Fermat's little theorem. It is still unclear whether the reverse circuit direction applies. So it is not known whether there are composite numbers with this property. After a score of 1994 would have such a number, more than 10,000 decimal places.

Giugas conjecture is equivalent to the following statement: No natural number is also Giuga and Carmichael number.

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

With the Bernoulli numbers.

Definition

A composite number is called Giuga number, if for all prime divisors of: tells.

The related to the Giuga numbers Carmichael numbers have a similar characterization: A composite number is called Carmichael number if and only if for all prime divisors of: tells.

Equivalent characterizations

The Giuga numbers can still characterize to other species: Be a composite number and the amount of prime divisors of. Then:

  • This number is accurate then a Giuga number if and only if.
  • This number is accurate then a Giuga number if and only if is square-free and
  • This number is accurate then a Giuga number if and only if.

It denotes the Euler φ - function and Bernoulli numbers.

Known Giuga numbers

  • Three factors: 30 = 2 * 3 * 5
  • 858 = 2 * 3 * 11 * 13
  • 1722 = 2 * 3 * 7 * 41
  • 66 198 = 2 * 3 * 11 * 17 * 59
  • 2,214,408,306 = 2 * 3 * 11 * 23 * 31 * 47,057
  • 24,423,128,562 = 2 * 3 * 7 * 43 * 3041 * 4447
  • 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
  • 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
  • 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 square-free.
  • 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. An odd Giuga number should consist of at least 14 prime factors. Since all Carmichael numbers are odd, even Giugas guess would be proved if one could prove that all Giuga numbers are.
Abundant number
266981
de