Fermat upside-down
The world-famous Fermat's Last Theorem states that the equation an+bn=cn has no integer solutions where n > 2. It was proposed by Pierre de Fermat in 1637, and was not proven until 1995, 358 years later, when Andrew Wiles proved it. It took over 100 pages for him to do this, and even then, he relied on others' work in elliptic curve theory over the previous forty years.
I have a much simpler theorem I want proved. Prove that the equation na+nb=nc has no integer solutions where n > 2.
I have a much simpler theorem I want proved. Prove that the equation na+nb=nc has no integer solutions where n > 2.
Labels: mathschallenge
posted by Ross at
10:05 PM
4 Comments:
This post has been removed by the author.
It's either too easy, or it's me who's missing something here.
Anyway, I came with the following:
Let b = d^a, and c = e^a
n^a + n^d^a = n^e^a
n^a + m^a = k^a
Which has no integer solution for a > 2 according to Fermat's.
Mohamed, it seems to me that by letting b and c equal that, you've restricted the domain. If a is 5, for example, then b and c can only be taken from the set of fifth powers of integers, not from all integers.
c must be bigger than a and b
first assume b is >= a
n^a+n^b=n^c can be simplified to
1+n^(b-a)=n^(c-a)
OR
n^(c-a)-n^(b-a)=1
the difference between exponents increase as n increases and as the c increases.
The only integer that has another exponent exactly 1 away from the previous term is 2. i.e. 2^0 and 2^1.
the smallest all integer difference between any exponent of an n will be n^1-n^0 or (n-1).
n-1=1 is only true if
n=2 and the equation must have (c-a)=1 and b-a=0
2^1-2^0=2-1=1
n>2 thus no solutions exist
Cam
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