What is congruent number theory?

What is congruent number theory?

In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle.

What is congruence theory?

As we mentioned in the introduction, the theory of congruences was developed by Gauss at the beginning of the nineteenth century. We say that a is congruent to b modulo m if m∣(a−b) where a and b are integers, i.e. if a=b+km where k∈Z. If a is congruent to b modulo m, we write a≡b(mod m).

What is congruence mod’m give an example?

Congruence modulo m divides the set ZZ of all integers into m subsets called residue classes. For example, if m = 2, then the two residue classes are the even integers and the odd integers. Congruence is symmetric, i.e., if a ≡ b (mod m), then b ≡ a (mod m).

What are congruent examples?

The word ‘congruent’ means ‘exactly equal’ in terms of shape and size. Even when we turn, flip, or rotate the shapes, they remain equal. For example, draw two circles of the same radius, then cut them out and place them on one another.

What is congruence mod?

Two integers, a and b, are congruent modulo n if and only if they have the same remainder when divided by n. In other words, for some integer k (positive or negative): a=b+kn. or. a−b=kn.

What is congruent in discrete mathematics?

If two numbers and have the property that their difference is integrally divisible by a number (i.e., is an integer), then and are said to be “congruent modulo .” The number is called the modulus, and the statement ” is congruent to (modulo )” is written mathematically as. (1)

How do you solve modulo problems?

How to calculate the modulo – an example

1. Start by choosing the initial number (before performing the modulo operation).
2. Choose the divisor.
3. Divide one number by the other, rounding down: 250 / 24 = 10 .
4. Multiply the divisor by the quotient.
5. Subtract this number from your initial number (dividend).

What is congruence relationship give an example?

Definition 3.1 If a and b are integers and n > 0, we write a ≡ b mod n to mean n|(b − a). We read this as “a is congruent to b modulo (or mod) n. For example, 29 ≡ 8 mod 7, and 60 ≡ 0 mod 15. The notation is used because the properties of congruence “≡” are very similar to the properties of equality “=”.

What are the applications of number theory?

The best known application of number theory is public key cryptography, such as the RSA algorithm. Public key cryptography in turn enables many technologies we take for granted, such as the ability to make secure online transactions.

What is the theory of numbers?

Number Theory. Number theory is a vast and fascinating field of mathematics, sometimes called “higher arithmetic,” consisting of the study of the properties of whole numbers.

What is the algebraic number theory?

Algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers,…

What is the definition of congruent in psychology?

In psychology and NLP , congruence could be defined as rapport within oneself, or internal and external consistency, perceived by others as sincerity or certainty.