In number theory , a quartic prime number (from the English quartan prime ) is a prime number of the form with integer and .
p
=
x
4th
+
y
4th
{\ displaystyle p = x ^ {4} + y ^ {4}}
x
>
0
{\ displaystyle x> 0}
y
>
0
{\ displaystyle y> 0}
Examples
The number is (the only even) quartic prime number.
2
=
1
+
1
=
1
4th
+
1
4th
{\ displaystyle 2 = 1 + 1 = 1 ^ {4} + 1 ^ {4}}
The number is a quartic prime.
97
=
16
+
81
=
2
4th
+
3
4th
{\ 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 )
p
=
919444
1048576
+
1
=
(
919444
262144
)
4th
+
1
4th
{\ 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.
6.253.210
{\ displaystyle 6.253.210}
properties
Let with an (odd) quartic prime number. Then:
p
∈
P
{\ displaystyle p \ in \ mathbb {P}}
p
>
2
{\ displaystyle p> 2}
p
=
16
n
+
1
{\ displaystyle p = 16n + 1}
With
n
∈
N
{\ displaystyle n \ in \ mathbb {N}}
In other words:
p
≡
1
(
mod
16
)
{\ displaystyle p \ equiv 1 {\ pmod {16}}}
Let with an (odd) quartic prime number. Then:
p
=
x
4th
+
y
4th
∈
P
{\ displaystyle p = x ^ {4} + y ^ {4} \ in \ mathbb {P}}
p
>
2
{\ displaystyle p> 2}
If is odd, must be even or vice versa.
x
{\ displaystyle x}
y
{\ 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 .
x
{\ displaystyle x}
y
{\ displaystyle y}
x
4th
{\ displaystyle x ^ {4}}
y
4th
{\ displaystyle y ^ {4}}
p
=
x
4th
+
y
4th
{\ displaystyle p = x ^ {4} + y ^ {4}}
p
>
2
{\ 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 .
x
{\ displaystyle x}
y
{\ displaystyle y}
x
4th
{\ displaystyle x ^ {4}}
y
4th
{\ displaystyle y ^ {4}}
p
=
x
4th
+
y
4th
{\ displaystyle p = x ^ {4} + y ^ {4}}
p
>
2
{\ displaystyle p> 2}
All that remains is that either or is odd and the other is even.
x
{\ displaystyle x}
y
{\ displaystyle y}
◻
{\ displaystyle \ Box}
See also
Individual evidence
↑ 919444 1048576 + 1 on Prime Pages
↑ 919444 1048576 + 1 on primegrid.com (PDF)
↑ AJC Cunningham: High quartan factorizations and primes. Messenger of Mathematics 36 , 1907, pp. 145–174 , accessed July 7, 2018 .
Web links
swell
Neil Sloane : A Handbook of Integer Sequences . Academic Press, Inc., New York 1973, ISBN 1-4832-4665-5 , pp. 205 .
formula based
Carol ((2 n - 1) 2 - 2) |
Cullen ( n ⋅2 n + 1) |
Double Mersenne (2 2 p - 1 - 1) |
Euclid ( p n # + 1) |
Factorial ( n! ± 1) |
Fermat (2 2 n + 1) |
Cubic ( x 3 - y 3 ) / ( x - y ) |
Kynea ((2 n + 1) 2 - 2) |
Leyland ( x y + y x ) |
Mersenne (2 p - 1) |
Mills ( A 3 n ) |
Pierpont (2 u ⋅3 v + 1) |
Primorial ( p n # ± 1) |
Proth ( k ⋅2 n + 1) |
Pythagorean (4 n + 1) |
Quartic ( x 4 + y 4 ) |
Thabit (3⋅2 n - 1) |
Wagstaff ((2 p + 1) / 3) |
Williams (( b-1 ) ⋅ b n - 1)
Woodall ( n ⋅2 n - 1)
Prime number follow
Bell |
Fibonacci |
Lucas |
Motzkin |
Pell |
Perrin
property-based
Elitist |
Fortunate |
Good |
Happy |
Higgs |
High quotient |
Isolated |
Pillai |
Ramanujan |
Regular |
Strong |
Star |
Wall – Sun – Sun |
Wieferich |
Wilson
base dependent
Belphegor |
Champernowne |
Dihedral |
Unique |
Happy |
Keith |
Long |
Minimal |
Mirp |
Permutable |
Primeval |
Palindrome |
Repunit ((10 n - 1) / 9) |
Weak |
Smarandache – Wellin |
Strictly non-palindromic |
Strobogrammatic |
Tetradic |
Trunkable |
circular
based on tuples
Balanced ( p - n , p , p + n) |
Chen |
Cousin ( p , p + 4) |
Cunningham ( p , 2 p ± 1, ...) |
Triplet ( p , p + 2 or p + 4, p + 6) |
Constellation |
Sexy ( p , p + 6) |
Safe ( p , ( p - 1) / 2) |
Sophie Germain ( p , 2 p + 1) |
Quadruplets ( p , p + 2, p + 6, p + 8) |
Twin ( p , p + 2) |
Twin bi-chain ( n ± 1, 2 n ± 1, ...)
according to size
Titanic (1,000+ digits) |
Gigantic (10,000+ digits) |
Mega (1,000,000+ digits) |
Beva (1,000,000,000+ positions)
Composed
Carmichael |
Euler's pseudo |
Almost |
Fermatsche pseudo |
Pseudo |
Semi |
Strong pseudo |
Super Euler's pseudo
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