Pythagorean prime is a prime number which can be denoted as 4n + 1, where n has to be a positive integer. All primes of this form can be represented as a sum of two squares (p = a2 + b2), e.g., 5 is a Pythagorean prime, as 5 = 4 × 1 + 1 and it can be written as 5 = 22 + 12.
These primes are named after the famous Pythagorean theorem due to their connection with representing a sum of two squares.
Pythagorean Prime Formula:
For every Pythagorean prime (p), it can be represented in the form:
p = 4n + 1
In other words, a prime number p is a Pythagorean prime if p ≡ 1 ( mod 4).
First few Pythagorean primes are:
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, . . .
Examples of Pythagorean prime
Here is a table showing Pythagorean primes in both the forms 4n + 1, as well as a sum of two squares:
Pythagorean Prime | Form 4n + 1 | Sum of Two Squares (a2 + b2) |
|---|---|---|
5 | 4 × 1 + 1 | 12 + 22 |
13 | 4 × 3 + 1 | 22 + 33 |
17 | 4 × 4 + 1 | 12 + 42 |
29 | 4 × 7 + 1 | 52 + 22 |
37 | 4 × 9 + 1 | 62 + 12 |
41 | 4 × 10 + 1 | 42 + 52 |
53 | 4 × 13 + 1 | 22 + 72 |
63 | 4 × 15 + 1 | 52 + 62 |
This table shows how each Pythagorean primes can be expressed in both the forms.
Conclusion
Pythagorean primes are special prime numbers that can be written as the sum of two squares and follow the form 4n+1. They are closely connected to the Pythagorean Theorem because they can represent the hypotenuse of a right triangle
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