Prove that if p,q,r and s are odd integers, then this equation has no integer roots:

x^10 + px^9 - qx^7 + rx^4 - s = 0

Printable View

- May 8th 2010, 08:03 PMthe undertakerNo integer solutions to x^10 + px^9 - qx^7 + rx^4 - s = 0
Prove that if p,q,r and s are odd integers, then this equation has no integer roots:

x^10 + px^9 - qx^7 + rx^4 - s = 0 - May 8th 2010, 09:05 PMSoroban
Hello, the undertaker!

I baby-talked my way through it . . .

Quote:

Prove that if are odd integers,

then: . .has no integer roots.

Note that:

An even integer raised to any positive integral power is even: .

An odd integer raised to any positive integral power is odd: .

We have: .

Suppose is even.

We have: .

. . . . . . . . .

And the difference of an even number and an odd number canbe zero.*not*

Suppose is odd.

We have: .

. . . . . . . . .

And the difference of an odd number and an even number canbe zero.*not*

- May 8th 2010, 10:59 PMBacterius
Geez, that was awesome Soroban. Such a simple technique for such a seemingly intractable problem ! Too bad I can't triple-thank you for this one.

The only improvement I can suggest is putting the "even" and "odd" tags between \mathrm{}, like .

Otherwise, (Clapping)