- #1
barryn56
- 12
- 0
Can someone point out the error in the following "proof":
Prove a^n + b^n =/ c^n for n>2, a,b,c>1 (=/ means not equal to)
Let b=xa where x>1 and is from the set of real numbers generated by fractions, such that b is an integer
so:
a^n + (xa)^n =/ c^n
Expanding
a^n + x^n.a^n =/ c^n
then, taking the common factor a^n out
a^n(1+x^n) =/ c^n
then dividing through by a^n
1+x^n =/ c^n/a^n
Substitute y from the set of real numbers given by the fraction c/a, then
1+x^n =/ y^n
or:
y^n - x^n =/ 1
Factorizing:
x^n - y^n = (x - y)*(x^[n-1] + x^[n-2]*y + ... + y^[n-1]).
If the left side equals 1, then x > y, and x - y must be a positive
divisor of 1, namely 1. Then x = y + 1. Substitute that into the
second factor above, and set that second factor also equal to 1. That
should give you a contradiction. The contradiction means that the
assumption that the left side equals 1 must be false, and then you're
done.
Prove a^n + b^n =/ c^n for n>2, a,b,c>1 (=/ means not equal to)
Let b=xa where x>1 and is from the set of real numbers generated by fractions, such that b is an integer
so:
a^n + (xa)^n =/ c^n
Expanding
a^n + x^n.a^n =/ c^n
then, taking the common factor a^n out
a^n(1+x^n) =/ c^n
then dividing through by a^n
1+x^n =/ c^n/a^n
Substitute y from the set of real numbers given by the fraction c/a, then
1+x^n =/ y^n
or:
y^n - x^n =/ 1
Factorizing:
x^n - y^n = (x - y)*(x^[n-1] + x^[n-2]*y + ... + y^[n-1]).
If the left side equals 1, then x > y, and x - y must be a positive
divisor of 1, namely 1. Then x = y + 1. Substitute that into the
second factor above, and set that second factor also equal to 1. That
should give you a contradiction. The contradiction means that the
assumption that the left side equals 1 must be false, and then you're
done.