# How to say that functions generate set members?

In a paper I am writing, I have seven abstract statements and, corresponding to the first one, I have a theorem statement that says I will prove that there is a Pythagorean triplet for every pair of natural numbers $$n,k$$ using functions: $$A(n,k)=(2n-1)^2+2(2n-1)k$$ $$B(n,k)=2(2n-1)k+2k^2$$ $$C(n,k)=(2n-1)^2+2(2n-1)k+2k^2$$ developed from $$my$$ observation that all $$interesting$$ triplets (especially primatives) are members of distinct sets as shown in the sample below (where $$n$$ is the set number and $$k$$ is the element number within the set). In each triplet in each set, the difference between B and C is always $$(2n-1)^2$$ and the increment between values of A is always $$2(2n-1)$$. The theorem statement so far is: $$\forall n,k\in\mathbb{N},∃ A,B,C\in\mathbb{N}|A^2+B^2=C^2\land \left\{A,B,C\right\}\in \left\{ ? \right\}$$ and here I want to correct the syntax for saying each triplet (A,B,C) is a unique element of a distinct set of triplets.

$$\begin{array}{c|c|c|c|c|} \text{Set_n}& \text{Triplet_1} & \text{Triplet_2} & \text{Triplet_3} & \text{Triplet_4}\\ \hline \text{Set_1} & 3,4,5 & 5,12,13& 7,24,25& 9,40,41\\ \hline \text{Set_2} & 15,8,17 & 21,20,29 &27,36,45 &33,56,65\\ \hline \text{Set_3} & 35,12,37 & 45,28,53 &55,48,73 &65,72,97 \\ \hline \text{Set_4} &63,16,65 &77,36,85 &91,60,109 &105,88,137\\ \hline \end{array}$$ I have written the proof(s) for the $$existence$$ and $$set$$-$$membership$$ but how do I describe the set membership in the theorem statement? In other words what goes inside the brackets with the question mark $$\left\{ ? \right\}$$ to show that every $$set$$ of triplets is an element of a greater set?

I am not sure that I correctly understood your question, but I guess that you are asking how to state that to distinct pairs $$(n,k)$$ of natural numbers correspond distinct triples $$(A(n,k),B(n,k),C(n,k))$$. That is, formally saying, that a map $$\Pi:\Bbb N\times \Bbb N$$, $$(n,k)\mapsto (A(n,k),B(n,k),C(n,k))$$ is injective. A different claim is that each Pythagorean triple $$(A,B,C)$$ equals $$(A(n,k),B(n,k),C(n,k))$$ for some natural $$n$$ and $$k$$. But if we are interested only in interesting triples then we don’t need this claim.

PS.

The theorem statement so far is: $$\forall n,k\in\mathbb{N},∃ A,B,C\in\mathbb{N}|A^2+B^2=C^2\land \left\{A,B,C\right\}\in \left\{ ? \right\}$$

When I was a schoolboy, I also tried to use such overformalized notation working with a close problems in order to be more cool. Now I know that it is better to formulate claims to be easy to read and understand, with more words, if needed. It is more professional, clear, and error-safe.

• The reason I want to include the unique set membership is because, without it, my theorem simply says there is a triplet for every pair of natural numbers. That part is well-known but my functions appear to be new because I have not been able to find anything like them in the literature or on the net in the past 10 years I have been working on this paper. I want to submit this to JAMS so I want it to be correct and to draw attention to the set properties. – poetasis Feb 23 at 12:57
• My email is poetasis@gmail.com. If you would care to see the paper, incomplete as it is, I can send you a pdf. – poetasis Feb 23 at 13:10
• My email is alexander.ravsky@uni-wuerzburg.de . You can send to me the paper, if you wish, but I have to confess that my knowledge of number theory is superficial, so I am not a right person to evaluate novelty and value of your results. I guess the most what I can do for you is to provide some general professional advices on paper-writing from a working mathematician, if you wish. But such advices probably are already contained in some known sources. Also I’m involved in many projects, so it is hard for me to find a free time, thus I do not guarantee that I’ll be able to read your paper. – Alex Ravsky Feb 24 at 6:18
• @poetasis Concerning a submission to JAMS. It has impact factor about 3, which is the highest value for AMS journals. I guess it means that only papers of high-level professionals can be published in it. For instance, I expect that my papers will be never published there. We published our paper, in which we solved a known problem which was opened for more than a half of a century, in PAMS, whose impact factor is only about 0.6. – Alex Ravsky Feb 24 at 6:19
• @poetasis I think you already defined the needed. If you want to group the triples then we can put straightforwardly $S_n=\{(A(n,k), B(n,k),C(n,k)):k\in\Bbb N\}$. – Alex Ravsky Feb 27 at 6:16

Give a definition:$$\forall n\in \Bbb Z^+ \,\;[\, S_n=\{(A,B,C)\in (\Bbb Z^+)^3\,: A^2+B^2=C^2\, \land\, C-B=(2n-1)^2\}\,].$$

Then you have a lemma: $$\forall n\in \Bbb Z^+\, (S_n\ne \emptyset).$$ Proof: If $$x\in \Bbb Z^+$$ and $$y=x+2n-1$$ then $$(y^2-x^2, 2xy,y^2+x^2)\in S_n.$$

It is a very ancient result that every primitive Pyth. triplet $$(A,B,C)$$ satisfies $$[\;\{A,B\}=\{y^2-x^2, 2xy\}\land z=y^2+x^2\; ]$$ for a unique $$(y,x)\in \Bbb Z^2....$$ and that $$x,y$$ are co-prime and that $$y-x$$ is odd.

• What does your answer have to do with my new functions? I especially want nothing to do with the $ancient$ result because my functions are far more useful, allowing me to find matching sides, matching perimeters, matching areas, and even create convex polygons, pyramids and so on using dissimilar triangles. – poetasis Feb 22 at 20:17