Quartic prime

In number theory , a quartic prime number (from the English quartan prime ) is a prime number of the form with integer and . ${\displaystyle p=x^{4}+y^{4}}$ ${\displaystyle x>0}$ ${\displaystyle y>0}$

Examples

• The number is (the only even) quartic prime number. ${\displaystyle 2=1+1=1^{4}+1^{4}}$
• The number is a quartic prime. ${\displaystyle 97=16+81=2^{4}+3^{4}}$
• The smallest quartic prime numbers are:

2, 17, 97, 257, 337, 641, 881, 1297, 2417, 2657, 3697, 4177, 4721, 6577, 10657, 12401, 14657, 14897, 15937, 16561, 28817, 38561, 39041, 49297, 54721, 65537, 65617, 66161, 66977, 80177, 83537, 83777, 89041, 105601, 107377, 119617, 121937, ... (sequence A002645 in OEIS )

• The currently largest known quartic prime (as of June 17, 2018) is also the largest known generalized Fermat prime number , namely

${\displaystyle p=919444^{1048576}+1=(919444^{262144})^{4}+1^{4}}$

She has jobs and was discovered by Sylvanus A. Zimmerman (USA) on August 29, 2017. ${\displaystyle 6.253.210}$

properties

• Let with an (odd) quartic prime number. Then: ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle p>2}$

${\displaystyle p=16n+1}$ With ${\displaystyle n\in \mathbb {N} }$

In other words:

${\displaystyle p\equiv 1{\pmod {16}}}$

• Let with an (odd) quartic prime number. Then: ${\displaystyle p=x^{4}+y^{4}\in \mathbb {P} }$ ${\displaystyle p>2}$

If is odd, must be even or vice versa. ${\displaystyle x}$ ${\displaystyle y}$

Proof:

Suppose both and are straight. Then there would also be and even and thus the sum of two even numbers would also be an even prime number. But this cannot be because of . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x^{4}}$ ${\displaystyle y^{4}}$ ${\displaystyle p=x^{4}+y^{4}}$ ${\displaystyle p>2}$

Suppose both and are odd. Then also and would be odd and thus the sum of two odd numbers would be an even prime number. But this cannot be because of . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x^{4}}$ ${\displaystyle y^{4}}$ ${\displaystyle p=x^{4}+y^{4}}$ ${\displaystyle p>2}$

All that remains is that either or is odd and the other is even. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \Box }$

See also

• Biquad

Individual evidence

1. ↑ 919444 ^1048576 + 1 on Prime Pages
2. ↑ 919444 ^1048576  + 1 on primegrid.com (PDF)
3. ↑ AJC Cunningham: High quartan factorizations and primes. Messenger of Mathematics 36 , 1907, pp. 145–174 , accessed July 7, 2018 .  

Web links

• Neil Sloane : A Handbook of Integer Sequences. Retrieved May 27, 2018 . 

swell

Neil Sloane : A Handbook of Integer Sequences . Academic Press, Inc., New York 1973, ISBN 1-4832-4665-5 , pp. 205 .