Math Is Fun Forum

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.

haha ty for the warm welcome
and yea, Im going tomorrow

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.