# Talk:Proof by infinite descent

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Field: Number theory

It is not true that infinite descent is a type of induction. If we want to prove by induction that every number x has property P, we must know that a specific number x1 has property P. From this we prove that every number has this property. On the other hand, If we want to prove by infinite descent that every number x has property P, we do NOT need to have a specific example for a number which has the property P. We just say that if there exists a number x that doesn't have property P, then there must exist a smaller number that doesn't have property P, and this is impossible - not because we know of a number x for which the property P is true, but because a descending sequence of natural numbers (say) is impossible.

No, infinite descent is no type of induction. It just means to show that there exists something smaller than the smallest one. I don't see, that this kind of proof has to do anything with infinity. --178.203.183.31 (talk) 17:32, 12 January 2012 (UTC)
If there existed an example with property P, then there would exist a smaller example with property P, and then there would exist a still smaller example with property P, and then there would exist a still smaller example with property P, .... Thus there is induction (infinitely repeated use of the same step) on a hypothetical starting point. The number of smaller and smaller examples in the sequence would be infinity, which is impossible because there's a finite number of integers smaller than any starting integer. Duoduoduo (talk) 18:14, 12 January 2012 (UTC)
Ok --178.203.183.31 (talk) 12:47, 13 January 2012 (UTC)

Actually this kind of proof uses the well-ordering principle of natural numbers. Accepting this principle is equivalent to accepting the principle of induction (they are equivalent). So in some way this is a proof by induction, since it uses the principle of induction. I believe there is proof of the equivalence in the Proof of mathematical induction article.

I think the proof could use some extra detail in the part which states that "3|a^2+b^2 <-> 3|a and 3|b". This can be done easily although I'm not comfortable with the syntax so maybe someone else can merge it as a reference note? (edit this page for reading the dem. since the line breaking gets messed up)

(1) Suppose 3|a, and 3 is prime 3|a -> 3|a^2 3|a^2+b^2 and 3|a^2 -> 3|b^2 Because 3 is prime, 3|b^2 -> 3|b So one possibility is 3|a and 3|b.

(2) Suppose not 3|a not 3|a -> not 3|a^2 3|a^2+b^2 and not 3|a^2 -> not 3|b^2 -> not 3|b Therefore 3 and b are coprimes, and 3 and a are coprimes, which we use in the congruence equations of fermat's little theorem: a^2 = a^(3-1) = 1 mod 3 b^2 = b^(3-1) = 1 mod 3 Then a^2 + b^2 = 1 + 1 mod 3 = 2 mod 3 and this is absurd. Finally 3 must divide a, and therefore divide b as proven in (1)

As far as the example above re: changing the proof, I didn't think that "n|a^2+b^2 <-> n|a and n|b" would need a proof here, maybe just a reference to a proof if even that. But for clarity, what comparison is used to say that one solution is less than another? No mention is made, which might be assuming more than it should be in this context.

## CAN I PUT THIS ON ?

If √2 is a rational number, then there must exist a set of integers {S1 S2 S3 ...} any one of which, when multiplied by √2 give an integer. This set must have a smallest member Let S1 be the smallest of this set. S1√2 = Integer. S1(√2 - 1) = an integer <S1 because (√2 - 1)< 1 so S1(√2 - 1)*√2 = an integer, so S1 cannot be the smallest integer with this property. I found this proof on the internet some time ago. I think it would serve as a very simple example under the heading : Simple application examples I did not invent this proof I found it here but this is my own explanation of it. So would I be OK as far as copyright is concerned, and can anyone see a mistake or room for improvement ? Robert2957 14:57, 28 October 2006 (UTC)

Ow... my eyes. The integers aren't well-ordered. It's necessary to make the set a subset of the positive integers (which are well-ordered) in order to assume that the set has a smalled member. Other than that, the notation hurts my eyes. —The preceding unsigned comment was added by 75.39.133.62 (talk) 07:16, 7 May 2007 (UTC).

The proof that 3|a^2+b^2 => 3|a and 3|b above I don't think you are right when you wrote b^2 = 1 (mod 3) in (2) because you have no guarantee that gcd(3,b)=1 to apply Fermat's little theorem. You have to break it into cases again, so say that not 3|b then follow what you wrote, else if 3|b then we have a clear contradiction. —Preceding unsigned comment added by 66.38.130.1 (talk) 21:30, 6 September 2007 (UTC)

This appears on the page, but it is not true:

Thus we have

   (3 a_2)^2 + (3 b_2)^2 = 3 \cdot (s_1^2+t_1^2)


and

   3(a_2^2+b_2^2) = s_1^2+t_1^2.\,


because you cannot factor out a 3 like that. Where a = 2 and b = 3 you would have:

Thus we have

   (3 2_2)^2 + (3 3_2)^2 = 3 \cdot (s_1^2+t_1^2)


and

   3(2_2^2+3_2^2) = s_1^2+t_1^2.\,  —Preceding unsigned comment added by 200.162.223.52 (talk) 16:40, 28 September 2007 (UTC)


## No valid examples of infinite descent in article

Neither proof in this article is a valid example of infinite descent. The first one assumes that the square root of 2 can be expressed as a fraction in lowest terms, and derives a contradiction. It is a proof by contradiction, but not a proof by infinite descent. The second proof assumes the existence of a minimal counterexample. This is a valid method, but it is not infinite descent.

The proofs should be rewritten to use infinite descent. For example, show that if (a_1, b_1) is a solution to the Diophantine equation (not necessarily a minimal solution) then there exists another solution (a_2, b_2) with a_2 < b_1. Or, show that if the square root of 2 can be expressed as p/q (not necessarily in lowest terms) then there exists another fraction with smaller denominator that expresses the square root of 2. -- Me 66.41.27.169 (talk) 18:57, 7 December 2008 (UTC)

I agree. Both proofs could be infinite descent if they were rewritten. For example, for the first one, instead of saying that there is a contradiction in p and q being coprime, say that you could infinitely divide both of them in half. For the second one, just rewrite so that there is no mention of a smallest member at the beginning. Just show that for any possible solution, there must be a smaller one, which is impossible for the set of positive integers. The exact same error exists over at Fundamental Theorem of Arithmetic Asmeurer (talkcontribs) 02:10, 21 January 2009 (UTC)

## Primitive Pythagorean triangles

The proof is nonsence, since the smaller primitive Pythagorean triangles don't have the desired properties. --178.203.183.31 (talk) 17:59, 8 January 2012 (UTC)

I assume that comment is in reference to the section Non-solvability of $r^2 + s^4 =t^4$. In fact the proof is impeccable, and equally importantly, it is referenced to a source. I recommend that you study the proof more carefully, maybe here or maybe in the cited source. But vandalizing it is inappropriate and can get you banned from Wikipedia. Duoduoduo (talk) 18:58, 8 January 2012 (UTC)
Indeed, the desired property of the side lenghts squared is $r^2 = s^4 + t^4$, meaning that two side lengths are squares. --178.203.183.31 (talk) 19:45, 8 January 2012 (UTC)
No, the section heading is Non-solvability of $r^2 + s^4 =t^4$. So one leg and the hypotenuse are squares. If you have further issues with the proof, you could think them through carefully and then submit them as a comment to the journal that published the proof. But a Wikipedia talk page is not the place for that. Duoduoduo (talk) 20:16, 8 January 2012 (UTC)
Yes, $r^2 + s^4 = t^4$ means that the legs and the hypotenuse are r,s², and t². So one leg and the hypotenuse are squares. The proof claims that there are smaller triangles having such properties. That's wrong. --178.203.183.31 (talk) 21:47, 8 January 2012 (UTC)

I've now made the text easier to read, in particular pointing out explicitly why the new hypotenuse in each case is smaller than the original.

I think I understand the source of your confusion. The premise to be disproven by infinite descent is stated as follows in the text:

It can be proven by proving that a Pythagorean triangle cannot have two sides each of which is each either a square or twice a square. Suppose there exists such a Pythagorean triangle.

In each of the three cases, we start with a triangle that has two sides (in the three cases respectively, y and z, y and x, or z and x) each of which is a square or twice a square, and in each case we end up with a smaller triangle that has two sides each of which is a square or twice a square. So we have proven something broader than what the section heading says: namely that $r^u + s^v =t^w$ is impossible with two of u, v, and w equaling 4 and the other one equaling 2. I'll clarify that in the text.Duoduoduo (talk) 22:51, 8 January 2012 (UTC)

== Someone please explain the | notation ==


As I write this, the article includes statements such as:

2 | p

and

$3 \mid a_1^2+b_1^2$

I've studied math through college level calculus but I'm not a mathematician. I feel like if I don't understand the notation used in the Wiki article, neither will most other people, particularly high school students who may be the best audience for such articles. I'm not suggesting not using the notation, but can someone please include a link that explains the notation for those of us who need help? This is particularly important with notation as it is very difficult to search on notation such as "|".

Jgro (talk) 21:19, 23 January 2012 (UTC)

Okay, I'll put an explanation in the article. Thanks for pointing this out! 2|p means that "2 divides p" -- that is, the integer p is divisible by 2. Duoduoduo (talk) 21:31, 23 January 2012 (UTC)
Thanks, Duoduoduo. While you're at it, could you add a step that explains why 2|p^2 implies 2|p? It's not true in the general case and how it is provably true because p is a natural number is not obvious to me. Jgro (talk) 21:46, 23 January 2012 (UTC)
If 2 does not divide p, then the prime factorization of p (the product of its primes) contains no 2's. So when you square p by squaring all its factors, there still are no 2's in the resulting prime factorization of p2. So since p2 has been found to be divisible by 2, p must be divisible by 2 as well. I'll put it in. Duoduoduo (talk) 02:01, 24 January 2012 (UTC)

NON-SOLVABILITY OF R^2 + S^4 = T^4

The case of y and z : a beautifull approach.

The case of y and x : if y is a square and x is not a square (for instance twice a square)we are lucky because then z is a square by assumption and we are back in case 1. However if y is a square and x = 2ab is a square we are in trouble because then not both a and b can be squares and the constructed PPT (y<sup">½, b, a) doesn't have two square sides.

A PPT (x, y, z) can not have 2 sides that both are twice a square, because then the other side would be devisable by 2 and hence gcd(x, y, z) ≥ 2. C.W. Vugs (talk) 11:15, 7 December 2012 (UTC)