Super Euler's pseudoprime number

A Super Euler's pseudoprime is an Euler's pseudoprime with the base a, all of the divisors of which consist exclusively of the 1, prime numbers , other Euler's pseudoprimes with the same base a and themselves. The definition is equivalent: Super-Euler's prime number is called a composite number if these two factors satisfy the equations for each decomposition into two factors m ₁ and m ₂ . Super Euler's pseudo-prime numbers based on base 2 are also called Super Chicken numbers. ${\displaystyle n=m_{1}*m_{2}}$ ${\displaystyle a^{m_{i}}-a\equiv 0\mod m_{i}}$

properties

All factors of a super-Euler pseudoprime, including 1 and the super-Euler pseudoprime, have the following property:

${\displaystyle a^{d}-a}$is divisible by d .

alternatively it can also be written like this:

${\displaystyle a^{d-1}-1}$is divisible by d .

example

294409 is a Super-Euler pseudoprime with base 2. Its factors are 1, 37, 73, 109, 2701, 4033, 7957 and 294409.

37, 73 and 109 are prime numbers, 2701, 4033 and 7957 are themselves Super Euler's pseudoprimes as the base.

Super Euler's pseudoprimes with 3 or more prime factors

It is relatively easy to construct a Super-Euler pseudoprime with base a with three prime factors. To do this, one has to find three Euler's pseudoprimes with base a , which together have three common prime factors. The product of these three prime numbers is then in turn an Euler's pseudo-prime number, and thus a super-Euler's prime number.

Super Chicken Numbers with 3 prime factors
Super chicken number	Factorization	Bases	Divider
1105	5 13 17	18, 21, 38, 47, 103, 118, 157 ...	1, 5, 13, 17, 65, 85, 221, 1105
1885	5 13 29	12, 57, 86, 99, 157, 278, 1032	1, 5, 13, 29, 65, 145, 377, 1885
3913	7 · 13 · 43	79	1, 7, 13, 43, 91, 301, 559, 3913
4505	5 17 53	242	1, 5, 17, 53, 85, 265, 901, 4505
7657	13 19 31	37, 191	1, 13, 19, 31, 247, 403, 589, 7657
294409	37 · 73 · 109	2	1, 37, 73, 109, 2701, 4033, 7957 and the like. 294409
1398101	23 · 89 · 683	2	1, 23, 89, 683, 2047, 15709, 60787 and the like 1398101
1549411	31 151 331	2	1, 31, 151, 331, 4681, 10261, 49981 and the like 1549411
1840357	43 127 337	2	1, 43, 127, 337, 5461, 14491, 42799 and the like. 1840357
12599233	97 · 193 · 673	2	1, 97, 193, 673, 18721, 65281, 129889 and the like. 12599233
13421773	53 · 157 · 1613	2	1, 53, 157, 1613, 8321, 85489, 253241 and the like. 13421773
15162941	59 233 1103	2	1, 59, 233, 1103, 13747, 65077, 256999 and the like. 15162941
15732721	97 · 241 · 673	2	1, 97, 241, 673, 23377, 65281, 162193 and the like. 15732721

Super Chicken Numbers with up to 7 prime factors can be obtained from the following four sets:

{103, 307, 2143, 2857, 6529, 11119, 131071}

{709, 2833, 3541, 12037, 31153, 174877, 184081}

{1861, 5581, 11161, 26041, 37201, 87421, 102301}

{6421, 12841, 51361, 57781, 115561, 192601, 205441}

_{They come from Gerard Michon}

So 1.118.863.200.025.063.181.061.994.266.818.401 = 6421 * 12841 * 51361 * 57781 * 115561 * 192601 * 205441 is a super chicken number with seven prime factors, the divisors of prime numbers, chicken numbers and super chicken Numbers (there are a total of 120 Chicken Numbers).

Slimmed-down super chicken numbers

If you waive the condition that the divisors of super chicken numbers must also include chicken numbers other than the super chicken number itself, you can also add the chicken numbers, which have only two prime factors.

The smallest Super Chicken number slimmed down in this way is 341 with the prime divisors 11 and 31.

Web links

• Numericana