# What are congruent numbers

## Congruences

In number theory two whole numbers are called a and bcongruent modulom with a natural number m if the difference (a − b) is an integral multiple of m. Different numbers that are congruent modulom can therefore be made to coincide by "shifting" by a multiple of the number m.

### Notations in number theory

The following notations are often used for the statement "a and b are congruent modulom":
a≡bmodm.
If two numbers are congruent modulo a number m, dividing by m results in the same remainder.

### Examples

• 3≡24mod7, because 7 divides -21 (= 3−24)
• −31≡11mod7, because 7 divides -42 (= −31−11)
• −15≡ − 64mod7, because 7 divides 49 (= −15 - (- 64))

### Residual classes

The set of all integers congruent to a (modulo m) is called the remainder class of a modulo m:
aˉ: = {x∈Z: a≡xmodm}
There are therefore exactly m residue classes (0ˉ, 1ˉ,… m − 1) modulo m.
The remainder classes modulo m form a ring, the so-called. Residual class ring. If m is a prime number, they form a field.
The theory of congruences was developed by Carl Friedrich Gauß in 1801 in his work Disquisitiones Arithmeticae justified.

The decisive criterion is beauty; There is no permanent place in this world for ugly mathematics.

Godfrey Harold Hardy

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