Jadhav Prime Quadratic Theorem

In Mathematics, Jadhav's Prime Quadratic Theorem is based on Algebra and Number Theory. Discovered by an Indian Mathematician Jyotiraditya Jadhav. Stating a condition over the value of $x$ in the quadratic equation $ax^2+bx+c$.

Theorem

It states that if a Quadratic Equation $ax^2+bx+c$ is divided by $x$ then it gives the answer as an Integer if and only if $x$ is equal to 1, Prime Factors and composite divisor of the constant $c$ .

Let the set of prime factors of constant term $c$ be represented as $p.f.[c]$ and the set of all composite divisor of $c$ be $d[c]$

$\frac{ax^2+bx+c}{x} \in Z$ Iff $x \in$ $p.f.[c] \bigcup d[c] \bigcup {1}$ where $a,b,c \in Z$.

Historical Note

Jyotiraditya Jadhav is a school-student and is always curious about numerical patterns which fall under the branch of Number Theory. He formulated many arithmetic based equations before too like Jadhav Theorem, Jadhav Triads, Jadhav Arithmetic Merging Equation and many more. While he was solving a question relating to Quadratic Equations he found out this numerical pattern and organized the theorem over it.

Proving "Jadhav Prime Quadratic Theorem"

Now let us take $\frac{ax^2+bx+c}{x}$ written as $\frac{x[ax+b]+c}{x}$

To cancel out $x$ from the denominator we need $x$ in numerator and to take $x$ as common from whole quadratic equation we need to have $c$ as a composite number made up as prime-factors with at least one factor as $x$ or in other words $c$ should be a multiple of $x$ and hence telling us $x$ should at least be a prime factor, composite divisor or 1 to give the answer as an Integer.

Hence Proving Jadhav Prime Quadratic Theorem.

Original Research paper can be found here on Issuu


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