Pythagorean prime
Encyclopedia
A Pythagorean prime is prime number
of the form 4n + 1. These are exactly the primes that can be the hypotenuse
of a Pythagorean triangle.
The first few Pythagorean primes are
Fermat's theorem on sums of two squares states that these primes can be represented as sums of two squares uniquely (up to order), and that no other primes can be represented this way, aside from 2=12+12. Thus these primes (and 2) occur as norms of Gaussian integers, while other primes do not.
The law of quadratic reciprocity
says that if p and q are distinct odd primes, at least one of which is Pythagorean, then
p is a quadratic residue mod q if and only if
q is a quadratic residue mod p; by contrast, if neither p nor q is Pythagorean, then p is a quadratic residue mod q if and only if q is not a quadratic residue mod p. −1 is a quadratic residue mod p if and only if p is a Pythagorean prime (or 2).
In the field Z/p with p a Pythagorean prime, the polynomial x^2 = -1 has two solutions.
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
of the form 4n + 1. These are exactly the primes that can be the hypotenuse
Hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the...
of a Pythagorean triangle.
The first few Pythagorean primes are
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, 4141 (number)41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...
, 5353 (number)53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....
, 6161 (number)61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....
, 7373 (number)73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...
, 89, 9797 (number)97 is the natural number following 96 and preceding 98.-In mathematics:97 is the 25th prime number , following 89 and preceding 101. 97 is a Proth prime as it is 3 × 25 + 1.The numbers 97, 907, 9007, 90007 and 900007 are happy primes...
, 101101 (number)101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...
, 109109 (number)109 is the natural number following 108 and preceding 110.-In mathematics:One hundred [and] nine is the 29th prime number, and also a Chen prime. The previous prime is 107, making them both twin primes...
, 113113 (number)113 is the natural number following 112 and preceding 114.-In mathematics:One hundred [and] thirteen is the 30th prime number, following 109 and preceding 127, a Sophie Germain prime, a Chen prime and a Proth prime as it is a prime number of the form 7 × 24 + 1...
, … .
Fermat's theorem on sums of two squares states that these primes can be represented as sums of two squares uniquely (up to order), and that no other primes can be represented this way, aside from 2=12+12. Thus these primes (and 2) occur as norms of Gaussian integers, while other primes do not.
The law of quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
says that if p and q are distinct odd primes, at least one of which is Pythagorean, then
p is a quadratic residue mod q if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
q is a quadratic residue mod p; by contrast, if neither p nor q is Pythagorean, then p is a quadratic residue mod q if and only if q is not a quadratic residue mod p. −1 is a quadratic residue mod p if and only if p is a Pythagorean prime (or 2).
In the field Z/p with p a Pythagorean prime, the polynomial x^2 = -1 has two solutions.