Let a, b and c be odd positive integers . Show that the quadratic equation 𝑎𝑥^2+𝑏𝑥+𝑐=0 has no rational solution.
41073
Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd
146126
Infinite positive integer solutions of the equation: $x^2+x+1=(y^2+y+1)(n^2+n+1)$
33314
How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?
59653
$x^2-y^2=2s$, s cannot be an odd integer
51804
How to show that a root of the equation $x (x+1)(x+2) ....... (x+2009) = c $ can have multiplicity at most 2?
30511
Monic $f(x)\in\Bbb Z[x]$ has no rational root if $f(0)\ \&\ f(1)$ odd [Parity Root Test, Modular Root Test]
134834
$n^2 + (n+1)^2 = m^3$ has no solution in the positive integers
144769
Distinct roots of $ax^2-bx+c=0$ in $(0,1)$, where $a,b,c\in \mathbb{Z}^+$
71361
Prove that $x^x+y^y=z^z$ doesn't have integer solutions
22800
Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
98408
Show that $x^2+y^2+z^2=999$ has no integer solutions
123159
Solutions of the diophantine equation $x+y+z=xyz$?
28274
Prove that $x^2 + y^2 = 3(z^2 + m^2)$ has no solutions in integer
59091
A quartic diophantine equation
12150
For what value of m that the equation $y^2 = x^3 + m$ has no integral solutions?
147442
Quadratic Diophantine Equation $x^2 + axy + y^2 = z^2$
47001
Cubic diophantine equation