Test question

[Nick]

Originally Posted by lukerichards

Here's the GCSE spec I teach:

http://filestore.aqa.org.uk/resource...00-SP-2015.PDF

and the Additional Maths GCSE:

https://www.ocr.org.uk/Images/457916...from-2018-.pdf

These summaries clearly show that all the mathematics being taught was either known already to Euclid or did not appear until the 17th century with Descartes, Leibniz and Newton (or later).
There was a gap of 2000 years in which no notable progress was made!

I see that everyone is taught the uniqueness of prime factorization.
Is it hard to convince your students of that or do they find it natural (perhaps even taking it for granted)?

[lukerichards]

Originally Posted by Nick

These summaries clearly show that all the mathematics being taught was either known already to Euclid or did not appear until the 17th century with Descartes, Leibniz and Newton (or later).
There was a gap of 2000 years in which no notable progress was made!

I see that everyone is taught the uniqueness of prime factorization.
Is it hard to convince your students of that or do they find it natural (perhaps even taking it for granted)?

Even the weakest of students I teach rarely struggle to find a prime factorisation. Conceptually most find the uniqueness of it to be a conceptual stretch. As in, I rarely get much push-back from it when I state it as a fact, but I think because it doesn't really resonate it doesn't really stick.

[Dr Sardonicus]

Originally Posted by Batalov

But it is important to use n=3 here and x=8, y=1.

Oh, I agree. Once you know that x - y divides xⁿ - yⁿ, you need to follow up with the corollary that if a > 1 and b > 1,

x^a - y^a divides x^ab - y^ab.

I note that setting y = 1 makes the algebra considerably simpler.

EDIT: I also recalled a simplified induction argument, inspired by working a specific numerical case. It uses the formula

xⁿ⁺¹ - yⁿ⁺¹ = x*(xⁿ - yⁿ) + yⁿ(x - y)

If Q_n denotes the quotient (xⁿ - yⁿ)/(x - y), this gives the simple formula

Q_n+1 = x*Q_n + yⁿ.

[LaurV]

Originally Posted by Dr Sardonicus

...mathematical induction...

oh? the easiest way to prove that is to multiply left side with x-y, and cancel out all terms, proving that by induction sounds to me like laborious futile work...

[Dr Sardonicus]

Originally Posted by LaurV

oh? the easiest way to prove that is to multiply left side with x-y, and cancel out all terms, proving that by induction sounds to me like laborious futile work...

Once you have the formula -- sure, it's easy to verify. And it is amusing to do the multiplication and watch the wholesale cancellation.

But the simplified induction proof ain't too shabby. Armed with the the verification for n = 1

Q₁ = (x¹ -y¹)/(x - y) = 1

and the recursion formula

Q_n+1 = x*Q_n + yⁿ

we can just "turn the crank" to get

Q₂ = x*1 + y¹ = x + y,

Q₃ = x*(x+y) + y² = x² + x*y + y², etc.

Again, the induction problem I recall was merely to establish divisibility; and it may have been a specific case, like showing that 19 divides 20ⁿ - 1 for every positive integer n. But the obvious proof for the specific case works, mutatis mutandis, for the algebraic version.