# Category Number Theory

## Quadratic Residues – I

Definition. For and , Then , we define say that is residue of nth-degree modulo m , If there exist integer solutions for – Otherwise, It is non-residue of nth-degree modulo m. For we call it as quadratic, cubic and bi-quadratic residues respectively. Theorem : There are quadratic residues in . Proof : Consider […]

## Exponent of a Prime

We define is is the exponent of p in the expansion of a. Thus by definition we directly have the following facts – 1. 2. 3. Equality when 4. 5. We also have the legendre’s formula – 6. This function gets to be seriously useful many more plus it is beautiful !! . Setting such […]

## Divisibility 2 – The Block Split Method

The Block Split Method To derive the test of divisibility for n (a natural number), say we have this relation – Let us take a natural number . When will ? – Interpret the number taken this way – $latex a_{l}a_{l-1}\hdots a_2a_1= x\cdot 10^r +y$ If, So we are done if the last congruence relation alone is checked. […]

## Divisibility 1

All natural numbers(composite or prime) have divisibility conditions. Let us derive some with some interesting ideas. For 11 Div. Condition : Difference of the the sum of alternately placed digits is either zero or an integer multiple of 11. Proof: if n is even. if n is odd. If n is even , If n is odd , Take […]