Pythagorean Prime (original) (raw)

Last Updated : 23 Jul, 2025

**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 theoremdue 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:

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

**Read More,

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