Linear Diophantine Equation

from the name it's clear that it's an equation of degree 1.

ax+by=g.c.d.(a,b)=d ...(I)

is also an linear Diophantine equation.According to

*Bézout's*identity if equation number (I) holds true.i.e. gcd(a,b)=d satisfies then, there should be an integer solution of this equation.It's true for more than two variables as well.

Lets take an example

14x+35y=7

As, gcd(14,35)=7 hence

*Bézout's*identity holds true.

Now we can verify it.(x,y)::(-2,1) is an integer solution set of this equation.

Fermat's Last Theorem

If n=2 this equation becomes x^2+y^2=z^2 nothing but Pythagorian triplets of (x,y,z). Infinite numbers of solution exist.

But if n>=3 no solution exists. This is known as Fermat's Last Theorem

Pell's Equation

this equation is known as Pell's equation, Where n is a

**non square integer**.Trivially x=1, y=0 is one of the solutions of this equation.But there could be other values ,till infinity, as well.

Now by squaring we can write

(x^2-ny^2)^2=1

^{}=>(x^2+ny^2)^2-n(2xy)^2=1

=>(2x^2+1)^2-n(2xy)^2=1 [If x^2-ny^2=-1]...(I)

or, (2x^2-1)^2-n(2xy)^2=1[If x^2-ny^2=1]....(II)

(I) and (II) are also another form of Pell's equation.

Erdős–Straus conjecture

For n>=2, 4/n can be expressed as the summation of three fractional numbers.

Wacław Franciszek Sierpiński generalized this equation and stated that,

for any positive

*k*there exists a number

*N*such that, for all

*n*≥

*N*, there exists a solution in positive integers to

*k*/

*n*=1/x+1/y+1/z

one of my problem is 3x+20y=42.find solutions

ReplyDeleteInfinite solution exists.

ReplyDeleteExplanation (1)

(x,y)=>(6,6),(2,9),(14,0),(18,-3),(22,-6)...

Explanation (II)

Equation of a straight line

Infinite solution possible.