History Of Mathematics

[HaloNoble6]

Next Saturday, the 6th of August, at 5:00PM EST, I will be posting another trivia quesiton. This time around, the question will involve the history of Mathematics. Namely, it touches upon Math in Ancient Greece. Hope to see you then.

[HaloNoble6]

As stated earlier, this round is based on Mathematics from the classical Greek era.

Rules: Don't start posting answers until 5:00PM EST; all answers must be from memory, no googling or other such searching; and you have as many tries as you wish. I will post a small writeup sometime tomorrow afternoon. Until then, points are at stake. Game on!

1. The set of numbers (a,b,c) that satisfy the following are called what and named after whom: Integer pairs (a,c) for which there is an integer b satisfying a² + b² = c². (10 pts.)

2. When and how were solutions (a,b,c) to the above expression first recorded in history? (20 pts.)

3. Who was the first to provide a general solution, rather than particular solutions, to the above expression? (10 pts. for author, 15 pts. for textual source, 20 pts. for year source was written)

4. What is the general solution? (30 pts.)

[greich]

From my days in physics I remember the most obvious example of integers that match the equation are a=3, b=4, c=5 since 9 + 16 = 25.

I seem to remember that it was Rene Fermat after which the famous theorem is named that there are no integers that match the more general equation to the nth power. Fermat's theorem was proved recently.

So in answer to question 3 I'm guessing Rene Fermat.

P.S. How do you do superscripts?

Edited August 7, 2005 by greich

[HaloNoble6]

Perhaps it is Fermat that stated that theorem, but the proof I'm asking for is the one for a general solution to the one with powers of 2.

In forums that allow html, I use the < sup > tag for superscripts.

[softwareNerd]

If I remember right, I had a text book that said if anyone could formulate the general solution to that they should write to the author. Unless I'm thinking of something else, I would say the solution must have been found after 1990.

[HaloNoble6]

Hint: the answers to 1 and 3 are Greek Mathematicians from the classical era.

[nimble]

Well it is called the pythagorean theorum, so my guess is pythagoreus. My spelling might be a bit off.

[HaloNoble6]

5 pts. to nimble for getting half of the answer to question 1.

[Dagny]

It's been awhile but I remember from geometry something about Euclid proving Pythagoras' theorem. He wrote Elements ...and nowadays everyone refers to it as Euclidian geometry. Then again, its been awhile.

[HaloNoble6]

25 points to Dagny for guessing the author of the proof and the source of the proof.

[softwareNerd]

I need to go back to school!

[HaloNoble6]

I'm extending the close of this round so that Rational_One and Hal, among others, can take a crack at it. Perhaps I will post my writeup after they see this thread, or when Monday afternoon comes by, which ever comes first.

[Hal]

I know almost nothing about Greek mathematics, but I'll have a guess:

1. The set of numbers (a,b,c) that satisfy the following are called what and named after whom: Integer pairs (a,c) for which there is an integer b satisfying a² + b² = c². (10 pts.)

Something to do with Pythagorus I assume... I'll guess that they are called either 'pythagorean pairs', or 'pythagorean numbers'.

2. When and how were solutions (a,b,c) to the above expression first recorded in history? (20 pts.)

I've no idea when pythagorus lived, but I'd guess around ~500BC. As for 'how', I assume it was by using right angled triangles.

3. Who was the first to provide a general solution, rather than particular solutions, to the above expression? (10 pts. for author, 15 pts. for textual source, 20 pts. for year source was written)

Diophantus in Arithmetica?

4. What is the general solution? (30 pts.)

I dont understand what you mean by the general solution - what is the actual question? Do you want a decision procedure for whether an arbitrary c can be decomposed into the sum of 2 primes, or a way of generating all the ordered paris (a, satisfying a^2 + b^2 = c^2?

If you mean the solution to a^n + b^n = c^n (ie Fermat's last theorem) then Andrew Wiles proved there are no triplets (a,b,c) satisfying this for n > 2.

Edited August 7, 2005 by Hal

[HaloNoble6]

The general solution to finding all integers (a,b,c) that satisify the pythagorian equation.

Great guess with Diophantus. He did come up with the general solution, but after Euclid.

[HaloNoble6]

Pythagoras lived around 500 BC, but Pythagorean triples were first recorded as far back as 1800 BC in Babylonia, inscribed on a clay tablet, known as Plimpton 322, which systematically listed a large number of integer pairs (a,c) for which there is an integer b satisfying Pythagoras' equation.

The problem of Pythagorean triples was considered interesting in other civilizations that are known to have possessed Pythagoras' theorem; China between 200 BC and 220 AD, and India between 500 BC and 200 BC.

The most complete understanding of the problem in ancient times was achieved by Greek mathematics between Euclid, around 300 BC, and Diophantus, around 250 AD.

Today we know the general formula for generating Pythagorean triples is:

a = (p² - q²)*r, b = 2*q*p*r, c = (p² + q²)*r.

It is easy to see that a² + b² = c² when a, b, c are given by these formulas, and a, b, c will be integers if p, q, r are.

Sources

1. Stillwell, John, Mathematics and Its History, 2002, Springer-Verlang New York, Inc.