Pythagorean prime number

In number theory , a Pythagorean prime number (from the English pythagorean prime ) is a prime number of the form with (not to be confused with Pythagorean number ). If a prime number is not a Pythagorean prime, it is called a non-Pythagorean prime . ${\displaystyle p\in \mathbb {N} }$ ${\displaystyle p=4n+1}$ ${\displaystyle n\in \mathbb {N} }$

Examples

• The smallest Pythagorean prime numbers are the following:

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617,… (sequence A002144 in OEIS )

properties

• Any Pythagorean prime can be represented as the sum of two squares.

Proof:

The proof follows directly from Fermatsche's theorem about the sum of two squares . This theorem is sometimes called Girard 's theorem .

Example:

${\displaystyle 41=4^{2}+5^{2}}$, , ... ${\displaystyle 101=1^{2}+10^{2}}$

• The reverse of the above property is also true:

If the sum of two squares is an odd prime number, then it is a Pythagorean prime number.

Proof:

The proof also follows directly from Fermatsche's theorem about the sum of two squares:

For the square of an even number with the following applies: . ${\displaystyle n:=2k}$ ${\displaystyle n,k\in \mathbb {Z} }$ ${\displaystyle n^{2}=(2k)^{2}=4k^{2}\equiv 0{\pmod {4}}}$

For the square of an odd number with the following applies: . ${\displaystyle m:=2k+1}$ ${\displaystyle m,k\in \mathbb {Z} }$ ${\displaystyle m^{2}=(2k+1)^{2}=4k^{2}+4k+1\equiv 1{\pmod {4}}}$

For odd prime numbers the following applies: (for Pythagorean prime numbers) or (for non-Pythagorean prime numbers). ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle p\equiv 1{\pmod {4}}}$ ${\displaystyle p\equiv 3{\pmod {4}}}$

For the above reasons , the sum of two square numbers is always , or , but never . If it is an odd prime number, all that remains is and that is exactly the Pythagorean prime numbers. ${\displaystyle \equiv 0{\pmod {4}}}$ ${\displaystyle \equiv 1{\pmod {4}}}$ ${\displaystyle \equiv 2{\pmod {4}}}$ ${\displaystyle \equiv 3{\pmod {4}}}$ ${\displaystyle \equiv 1{\pmod {4}}}$ ${\displaystyle \Box }$

• For every Pythagorean prime number there is a right-angled triangle with hypotenuse length , which has integer leg lengths . ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle {\sqrt {p}}}$

The Pythagorean prime number and its square root as hypotenuses of right triangles and how to calculate the large one from the small triangle ${\displaystyle p=5}$

Proof:

See Pythagorean theorem

• If the prime number is the hypotenuse length of a right triangle, then it is a Pythagorean prime number and the largest part of a Pythagorean triple . ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle p}$
• There are an infinite number of Pythagorean prime numbers.

Proof:

See Dirichletscher prime number theorem

The primary race between 4n + 1 and 4n + 3

Be . Then: ${\displaystyle x\in \mathbb {N} }$

The number of Pythagorean prime numbers (of the form ) to is approximately the same as the number of non-Pythagorean prime numbers (of the form ) to . In particular, the number of Pythagorean prime numbers is often somewhat smaller. This phenomenon is called Chebyshev's bias ( en ) in English and comes from the mathematician Pafnuti Lwowitsch Chebyshev . ${\displaystyle 4n+1}$ ${\displaystyle x}$ ${\displaystyle 4n+3}$ ${\displaystyle x}$ ${\displaystyle x}$

Examples

• Until there are only two numbers among which there are more Pythagorean primes (of the form ) than non-Pythagorean (odd) primes (of the form ), namely and . Between and there are the same number and from there are again more non-Pythagorean (odd) prime numbers. ${\displaystyle x=600000}$ ${\displaystyle 4n+1}$ ${\displaystyle 4n+3}$ ${\displaystyle 26861}$ ${\displaystyle 26862}$ ${\displaystyle 26863}$ ${\displaystyle 26878}$ ${\displaystyle 26879}$
• The following list shows when a "change of leadership" takes place in the "race" Pythagorean primes against non-Pythagorean (odd) prime numbers (in English Where prime race 4n-1 vs. 4n + 1 changes leader ):

3, 26861, 26879, 616841, 617039, 617269, 617471, 617521, 617587, 617689, 617723, 622813, 623387, 623401, 623851, 623933, 624031, 624097, 624191, 624241, 627353, 62739, 6273569 627449, 627511, 627733, 627919, 628013, 628427, 628937, 629371, ... (sequence A007350 in OEIS )

Relation to Gaussian prime numbers

The norm of a Gaussian number of form is . The following applies: ${\displaystyle x+\mathrm {i} \cdot y}$ ${\displaystyle \|x+\mathrm {i} \cdot y\|=x^{2}+y^{2}}$

• A Pythagorean prime number (including the prime number ) can always be represented as the norm of a Gaussian integer. Odd non-Pythagorean prime numbers cannot. ${\displaystyle p=2}$
• A Pythagorean prime number is not a prime number in the set of Gaussian prime numbers. The real part and the imaginary part of their prime factors in this factorization are the leg lengths of the right triangle with a given hypotenuse length . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle p}$

Proof:

Any Pythagorean prime number can be broken down into . ${\displaystyle p=x^{2}+y^{2}}$ ${\displaystyle p=x^{2}+y^{2}=(x+\mathrm {i} \cdot y)\cdot (x-\mathrm {i} \cdot y)}$

• Compare also the group of rational points on the unit circle .

Square leftovers

• Let be two different odd prime numbers, where at least one of the two should be a Pythagorean prime number. Then: ${\displaystyle p,q\in \mathbb {P} }$

${\displaystyle p}$is quadratic remainder modulo if and only if quadratic remainder is modulo . ${\displaystyle q}$ ${\displaystyle q}$ ${\displaystyle p}$

In other words:

Be with and . Then with the Legendre symbol : ${\displaystyle p,q\in \mathbb {P} }$ ${\displaystyle p>2,q>2,p\not =q}$ ${\displaystyle p\equiv 1{\pmod {4}}}$

${\displaystyle \left({\frac {p}{q}}\right)=1\Longleftrightarrow \left({\frac {q}{p}}\right)=1}$

Example:

Be and . Then and thus the quadratic remainder is modulo . Conversely, and thus the quadratic remainder is modulo . ${\displaystyle p=37\equiv 1{\pmod {4}}}$ ${\displaystyle q=47\equiv 3{\pmod {4}}}$ ${\displaystyle 15^{2}=225\equiv 37{\pmod {47}}}$ ${\displaystyle 37}$ ${\displaystyle 47}$ ${\displaystyle 11^{2}=121\equiv 84\equiv 47\equiv 10{\pmod {37}}}$ ${\displaystyle 47}$ ${\displaystyle 37}$

• Let be two different odd prime numbers, where both should be non-Pythagorean prime numbers. Then: ${\displaystyle p,q\in \mathbb {P} }$

${\displaystyle p}$is a quadratic remainder modulo if and only if there is no quadratic remainder modulo . ${\displaystyle q}$ ${\displaystyle q}$ ${\displaystyle p}$

In other words:

Be with and . Then: ${\displaystyle p,q\in \mathbb {P} }$ ${\displaystyle p>2,q>2,p\not =q}$ ${\displaystyle p,q\equiv 3{\pmod {4}}}$

${\displaystyle \left({\frac {p}{q}}\right)=1\Longleftrightarrow \left({\frac {q}{p}}\right)=-1}$

Example:

Be and . Then and thus the quadratic remainder is modulo . Conversely, however, there is no with and therefore no quadratic remainder is modulo . ${\displaystyle p=47\equiv 3{\pmod {4}}}$ ${\displaystyle q=43\equiv 3{\pmod {4}}}$ ${\displaystyle 41^{2}=1681\equiv 47\equiv 4{\pmod {43}}}$ ${\displaystyle 47}$ ${\displaystyle 43}$ ${\displaystyle x}$ ${\displaystyle x^{2}\equiv 43{\pmod {47}}}$ ${\displaystyle 43}$ ${\displaystyle 47}$

See also

Quadratic law of reciprocity

Individual evidence

1. ↑ For the notation: In the current Duden - The great dictionary of the German language in ten volumes - ISBN 3-411-70360-1 , the adjective "Pythagorean" given in this notation and "Pythagor spelling ä isch" called Austrian special form.
2. ^ Leonard Eugene Dickson : History of the Theory of Numbers, Volume II, Diophantine Analysis, Chapter VI. Carnegie Institution of Washington, Publication No. 256, Vol. II, p. 228 , accessed on June 28, 2018 (English). 
3. ^ John Stillwell : Elements of Number Theory. Undergraduate Texts in Mathematics , 2003, p. 112 , accessed on June 28, 2018 (English). 
4. ↑ Eric W. Weisstein : Chebyshev Bias . In: MathWorld (English).
5. ↑ Michael Rubinstein, Peter Sarnak: Chebyshev's bias. Experimental Mathematics 3 (3), 1994, pp. 173-197 , accessed June 28, 2018 . 
6. ↑ ^{a b} https://www.math.uni-bielefeld.de/~sek/number/leit04.pdf Bielefeld University

Web links

• Laurence Eaves: 5, 13 and 137 are Pythagorean Primes. Numberphile, accessed June 27, 2018 . 
• Pythagorean Prime . In: PlanetMath . (English)