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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?

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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.

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  • $\begingroup$ 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. $\endgroup$ – poetasis Feb 23 at 12:57
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    $\begingroup$ My email is poetasis@gmail.com. If you would care to see the paper, incomplete as it is, I can send you a pdf. $\endgroup$ – poetasis Feb 23 at 13:10
  • $\begingroup$ 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. $\endgroup$ – Alex Ravsky Feb 24 at 6:18
  • $\begingroup$ @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. $\endgroup$ – Alex Ravsky Feb 24 at 6:19
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    $\begingroup$ @poetasis I think you already defined the needed. If you want to group the triples then we can put straightforwardly $S_n=\{(A(n,k), B(n,k),C(n,k)):k\in\Bbb N\}$. $\endgroup$ – Alex Ravsky Feb 27 at 6:16
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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.

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  • $\begingroup$ 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. $\endgroup$ – poetasis Feb 22 at 20:17

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