Talk:Freshman's dream

Examples

Hello everyone; I want one example for that? —Preceding unsigned comment added by 212.138.113.11 (talk) 08:06, 10 January 2009 (UTC)

What did you mean by that?

In this case, the "mistake" actually gives the correct result, due to p dividing all the binomial coefficients save the first and the last. This theorem demonstrates that exponentiation by p produces an endomorphism, known as the Frobenius endomorphism of the ring.

Does this mean that (x+y)^p = x^p+y^p in a special case? If so, what is it? —Preceding unsigned comment added by 212.138.69.18 (talk) 10:10, 10 June 2009 (UTC)

Yes, this means (x+y)^p = x^p+y^p in the special case stated in the sentence immediately preceding what you quoted: for a prime number p, if x and y are members of a commutative ring of characteristic p, then (x+y)^p = x^p+y^p. The special case concerns commutative rings of characteristic p. This does not concern the integers, rational numbers or the real numbers, as those all have characteristic 0. See the page on characteristic, specifically the section on rings. -Krasnoludek (talk) 16:54, 10 June 2009 (UTC)

I have to wonder which freshmen, exactly, make this mistake. High school freshmen are usually taking first year algebra or geometry, which usually do not cover exponents to a large enough extent to make students think that much about (x+y)^p, except for p=2. College freshmen are usually well past this point, having learned about the binomial theorem a long time ago (probably in second year algebra).

Of course, you have your exceptions. But regarding the predominant stage that freshmen are at, I don't think "freshman's dream" is a very appropriate name. I'm not arguing that the article's title should be changed; if society has chosen to name this error as the freshman's dream, then Wikipedia should reflect that. But, honestly, is the name really appropriate? --69.91.95.139 (talk) 15:54, 19 June 2009 (UTC)

It's not, and society often makes bad choices. --93.136.168.8 (talk) 15:32, 4 July 2009 (UTC)

Does the prime characteristic case ever really see any use, though?(x+y)^p = x+y = x^p+y^p in a ring of characteristic p, where p is prime, so I can't see why anyone would simplify (x+y)^p to x^p+y^p and not go all the way to just saying it's x+y. 206.76.109.184 (talk) 22:57, 8 June 2012 (UTC)

History of the term

Is there any history to this term that could be included in the article? Jwyrwas (talk) 23:26, 5 September 2009 (UTC)

Yes, there's a nice history to the term, which I'm now including in the article. Thanks for the suggestion! -Krasnoludek (talk) 12:00, 15 September 2009 (UTC)

Done. -Krasnoludek (talk) 12:33, 15 September 2009 (UTC)

Child's Binomial Theorem

However, the term "Child's Binomial Theorem" is also used to refer to the related theorem that a number n is prime iff $(x+1)^{n}\equiv x^{n}+1\!\!\mod n$ in the polynomial ring $\mathbb {Z} _{n}[x]$

This is incorrect, as you can see from the very source you cited! Reread section 2.2 in Granville's paper - he clearly states equation 2.2 ($(x+1)^{n}=x^{n}+1\!\!\mod n$) as "called, by some, the Child's Binomial Theorem". The equation you stated above is a direct consequence of Fermat's Little Theorem and it is also a rephrasing of the second paragraph ("The name "freshman's dream" also sometimes refers to...") of the introduction.

For these reasons I've removed this sentence but if someone wants to put in a new section for it, it could then say something along the lines of The related theorem that a number n is prime iff $(x+1)^{n}\equiv x^{n}+1\!\!\mod n$ in the polynomial ring $\mathbb {Z} _{n}[x]$ is a direct consequence of Fermat's Little Theorem and it is a key fact in modern primality testing.

Source for this would again be Granville's paper. 82.0.25.163 (talk) 17:36, 4 May 2010 (UTC)

Thanks for the correction. I had never heard the term "Child's Binomial Theorem" before and was looking for a legitimate source to attribute it to. Apparently I skimmed Granville's paper too quickly to notice which statement he was refering to as the Child's Binomial Theorem. I think, following your suggestion, I'll add the related theorem because it's relevant to the general topic of when this identity is true. -Krasnoludek (talk) 18:38, 5 May 2010 (UTC)

Other meaning of freshman's dream

I've also heard about another theorem called freshman's dream: in a non-Archimedean complete field, a series converges iff the sequence of terms converges. However, in the whole web I found only a single reference to this meaning: in the book by Cohn: "Basic algebra: groups, rings, and fields", p. 322, Exercise 4 at the end of Sect. 9.2.

128.176.181.5 (talk) 14:44, 9 January 2012 (UTC)