I was in Oxford on the weekend. It was just some slum-like dredges. Boring.
did you go into the city centre? its actually really nice
if you are a fan of academe then it has the most complete library in the uk (not sure about the world)
if you are not a fan then there is no point going.
...didn't notice this and started posting straight away. =p
I guess I should introduce myself properly, my names Matt and I'm currently working towards Maths, Further Maths, Physics and English Literature A-levels. I am taking mathematics at university and am going to Oxford for interview on Sunday
I am currently working on absorbing as much interesting maths as possible =p.
Leonard Euler was the first to prove that this equation has only one solution, y=5 and x=3. Thus, he proved that 26 is the only number among the infinity of numbers which is jammed between a perfect square and a perfect cube.
The proof is lengthy and sophisticated even for the present day mathematicians.
The Prince of Mathematicians, Euler, who is belived to have contributed more to mathematics than any other individual was also the first to make amajore breakthrough in proving Fermat's Last theorem. He had proved that
a³+b³ = c³ has no solution where a,b, and c are whole numbers, the only exception being a=b=c=0.
Generations of mathematicians later proved Fermat's Last theorem for certain prime number values of n. It was only during the fag end of the twentieth century that Andrew Wiles finally succeeded in proving Fermat's Last Theorem.
It was Fermat who proved that 26 is the only number between a perfect square and cube.
Also, the proof that a³ + b³ = c³ has no solutions was only a tiny part of the solution. Find a piece of evidence in an infinity to support a hypothesis (regardless of how hard it was to find) is useless (this has been demonstrated by the Riemann Hypothesis in which millions of zeroes have been found upon Riemanns critical line, not furthering research particularly).
First off, hiya .
Second, 26 is in fact the only number which can be written as a square plus one, cube minus one. This was proven by Fermat.
Thirdly, regarding the outputs of computers being considered proof, the four colour theorem (mentioned earlier) is a good example. The solution is frowned upon as aesthetics is always desired in proof. Unfortunately the approach used on the four colour theorem required testing thousands of possible combinations once the original map had been reduced. An interesting note is that Turing proved (indirectly) that a machine (human or mechanical) will never be able to decide whether any given theorem is provable or not (this was one of Hilberts problems proposed in 1900 and ties in almost directly with Godels incompleteness theorem).
Finally, the proof that I find most interesting is Liebniz's infinite sequence converging to pi/4 (I have no real notation software and I am new to the forums so I am not sure what everything means =p so let (b,a)∫f(x) dx be the integral of f(x) with respect to x between a and b.
Basic calculus shows:
arctan(b) = (b,0)∫[1/(1+x^2)]dx
It follows that
arctan(1) = (1,0)∫[1/(1+x^2)]dx = pi/4
Now look at the formula for the sum to infinity of a geometric progression:
1+q+q^2+q^3.... = 1/(1-q)
By inspection it is clear that:
1/(1+x^2) = 1-x^2+x^4-x^6...
Integrating this new expression between 1 and 0 term by term gives:
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9....
I just thought it was rather elegant.