%0 Journal Article
%T Tree of Fermat-Pramanik Series and Solution of <i>A<sup>M</sup> </i>+<i>B</i><sup>2</sup> =<i>C</i><sup>2</sup> with Integers Produces a New Series of  (<i>C</i><sub>1</sub><sup>2</sup>- <i>B</i><sub>1</sub><sup>2</sup>)=(<i>C</i><sub>2</sub><sup>2</sup>- <i>B</i><sub>2</sub><sup>2</sup>)=(<i>C</i><sub>3</sub><sup>2</sup>- <i>B</i><sub>3</sub><sup>2</sup>)=Others
%A Panchanan Pramanik
%A Susmita Pramanik
%A Sabyasachi Sen
%J Advances in Pure Mathematics
%P 160-166
%@ 2160-0384
%D 2024
%I Scientific Research Publishing
%R 10.4236/apm.2024.143008
%X <p>
	The Fermat¨CPramanik series are like below:
</p>
<p>
	<img src=\"Edit_ddb18082-3f71-417d-8155-7eb6b957e731.png\" width=\"314\" height=\"25\" alt=\"\" />.<span style=\"line-height:1.5;\">The mathematical principle has been established by factorization principle. The Fermat-Pramanik tree can be grown. It produces branched Fermat-Pramanik series using same principle making Fermat-Pramanik chain. Branched chain can be propagated at any point of the main chain with indefinite length using factorization principle as follows: </span> 
</p>
<p>
	<span style=\"line-height:1.5;\"><img src=\"Edit_19d0c28d-a92e-402d-94bc-0ad15989fe1a.png\" width=\"368\" height=\"70\" alt=\"\" /><br />
</span> 
</p>
<p>
	<span style=\"line-height:1.5;\">Same principle is applicable for integer solutions of <em>A</em><sup><em>M</em></sup>+<em>B</em><sup>2</sup>=<em>C</em><sup>2</sup>which produces series of the type <img src=\"Edit_debdab6b-80af-486e-af48-64c787f26bbf.png\" width=\"310\" height=\"25\" alt=\"\" />. It has been shown that this equation is solvable with <em>N</em><span style=\"white-space:nowrap;\">{</span><em>A, B, C, M<span style=\"white-space:nowrap;\">}</span></em>. <img src=\"Edit_80122164-1305-403b-80d9-2f2c62bee23e.png\" width=\"333\" height=\"21\" alt=\"\" />where <img src=\"Edit_632ec2ed-e525-469a-851c-982254d571da.png\" width=\"77\" height=\"21\" alt=\"\" />, <img src=\"Edit_e5b23d21-f593-470c-bb57-f92921ce627f.png\" width=\"91\" height=\"21\" alt=\"\" />, <em>M</em>=<em>M</em><sub>1</sub>+<em>M</em><sub>2</sub> and <em>M</em><sub>1</sub><span style=\"white-space:nowrap;\">&gt;</span><em>M</em><sub>2</sub>. Subsequently, it has been shown that <img src=\"Edit_2e385d4a-87e7-4ec3-9c67-1758773e4b95.png\" width=\"551\" height=\"24\" alt=\"\" />using <span style=\"white-space:nowrap;\"><em>M</em>= <em>M</em><sub>1</sub>+<em>M</em><sub>2</sub>+<em>M</em><sub>3</sub>+..</span>. The combinations of <em>M</em>s should be taken so that the values of both the parts <span style=\"white-space:nowrap;\">(<em>C</em><sub><em>n</em></sub>+<em>B</em></span><sub><em>n</em></sub>)&nbsp;and <span style=\"white-space:nowrap;\">(<em>C<sub>n</sub></em>-<em>B<sub>n</sub></em>) </span>should be even or odd for obtaining <em>Z</em><span style=\"white-space:nowrap;\">{</span><em>B</em>,<em>C<span style=\"white-space:nowrap;\">}</span></em>. Hence, it has been shown that the Fermat triple can generate a) Fermat-Pramanik multiplate, b) Fermat-Pramanik Branched multiplate and c) Fermat-Pramanik deductive series. All these formalisms are useful for development of new principle of
%K Fermat Theorem
%K Fermat-Pramanik Tree
%K Solution of &lt
%K i&gt
%K A&lt
%K sup&gt
%K M&lt
%K /sup&gt
%K &lt
%K /i&gt
%K +&lt
%K i&gt
%K B&lt
%K /i&gt
%K &lt
%K sup&gt
%K 2&lt
%K /sup&gt
%K =&lt
%K i&gt
%K C&lt
%K /i&gt
%K &lt
%K sup&gt
%K 2&lt
%K /sup&gt
%K 
%K Deductive Series
%K Generation of FermatˇŻs Triode
%K Generation of Fermat Series
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=131887