Fermat's Last Theorem: A Combinatorial Form
Keith Raskin ???? ???? ????
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Keith Raskin ???? ???? ????
Math Teacher, Author: Math, Humor, Fiction, Essay; FOLLOW ONLY, 30K connections ??
For what it's worth, this is a combinatorial form of F.L.T.
x^n = Sum (m = 1 to n) [Sum (r = 1 to m) [ nCm mCr x^(n – m) a^(m – r) b^r ] ]
is never true for positive integers n (greater than two), x, a and b. The equation is equivalent to x^n + (x + a)^n = (x + a + b)^n.
Of course, when n = 2 the solutions are all of the Pythagorean triples, and the equation has the form x^2 = 2xb + 2ab + b^2.
Math Teacher, Author: Math, Humor, Fiction, Essay; FOLLOW ONLY, 30K connections ??
7 年Yes, but students must first care about the question, must be curious or made curious to some extent about why there are infinite solutions (all of the Pythagorean triples, also areas of squares) at 2, yet no solutions at 3 (for the volumes of cubes) or beyond (the volumes of hypercubes). And of course the whole history of Fermat's margin and all of the interest, and even the temporary gap in Wiles' proof can be tantalizing, yet for some soporific.
Retired Educator : Leadership Experience in Public and Private Sector
7 年I like this a lot. I think the most significant thing though to ask your students is the “Why” . Why do we know it won’t work for integers greater than 2? Why is this historical discovery important in making connections in mathematics? The key to engaging anyone to learn is questioning. And it truly supports my theory that we should be teaching young children the history behind mathematics so they can make solid connections in number theory. The true power of teaching is getting students to question and explain their thinking and make connections.