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How to generate a rigorous proof for - The specific conjecture which states that we can always frame every Prime number in an exponentially framed equation with perfect cubes & squares. I'll present the respective segments of the conjecture and its Prototype.

Conjecture's statement -

The square of every Prime number can be expressed as a sum or difference of some perfect cube and any another Prime

Mathematical Structure $a^3 \pm P_{k} = P^2_n$ , $a,n, k\in\mathbb{N}$ , Where $P_m$ denotes the $m$th Prime respectively .

Progress on Initial inputs and outputs-

$\begin{array}{||c || c ||} \hline a^3 \pm p_k& p_n^2 \\ \hline 1+3& 2^2\\ \hline 512-503& 3^2\\ \hline 8+17& 5^2\\ \hline 8+41& 7^2\\ \hline 8+113& 11^2\\ \hline 216-47& 13^2\\ \hline 8+281& 17^2\\ \hline 8+353& 19^2\\ \hline 8+521& 23^2\\ \hline 1728-887& 29^2\\ \hline 8+953& 31^2\\ \hline 8+1361& 37^2\\ \hline 1728-47& 41^2\\ \hline 8000-6151& 43^2\\ \hline 216+1993& 47^2\\ \hline 8+2801& 53^2\\ \hline 512+2969& 59^2\\ \hline 512+3209& 61^2\\ \hline 8+4481& 67^2\\ \hline 1728+3313& 71^2\\ \hline 216+5113& 73^2\\ \hline 1728+4513& 79^2\\ \hline 216+6673& 83^2\\ \hline 8000-79& 89^2\\ \hline ...\pm...& 97^2\\ \hline ...\pm...& 101^2\\ \hline ...\pm...& 103^2\\ \hline \end{array}$

Conjecture's Prototype

The sum or difference of square of every Prime Number with some Perfect cube yields any another Prime number.

Mathematical Structure $(m^3 \pm P^2_{b} = P_c)$, $m,b,c \in\mathbb{N}$ and $P_n$ is the nth Prime Number respectively.

The above parent equation can be framed in other $3$ ways aslike -

$P_c - m^3 = \pm P^2_b$ which clearly generates the expression for square of every Prime number as per the conjecture constraints.

$|P_c - m^3| = |P^2_b|$ where taking modulus yields positive results, $P_c \mp P^2_b =m^3$

Initial inputs and outputs-

$\begin{array}{||c || c ||} \hline P_c - m^3 & \pm P_b^2 \\ \hline\hline 31-27& 2^2\\ \hline 17-8& 3^2\\ \hline 89-64& 5^2\\ \hline 113-64& 7^2\\ \hline 337-216& 11^2\\ \hline 233-64& 13^2\\ \hline 353-64& 17^2\\ \hline 577-216& 19^2\\ \hline 593-64& 23^2\\ \hline 887-1728& -29^2\\ \hline 2689-1728& 31^2\\ \hline 1433-64& 37^2\\ \hline 1063-2744& -41^2\\ \hline 1913-64& 43^2\\ \hline 2273-64& 47^2\\ \hline 3023-5832& -53^2\\ \hline 3697-216& 59^2\\ \hline 5449-1728& 61^2\\ \hline 6217-1728& 67^2\\ \hline 8783-13824& -71^2\\ \hline 5393-64& 73^2\\ \hline \end{array}$

The parameter $a$ in Primary conjecture and the parameter $m$ in its prototype takes infinitely many but unique values .

In order to prove it , one needs to overcome a new level of challenges it brings in Number theory , So far many efforts have been made to prove the Conjecture but we couldn't prove it for every value of $a$ & $m$ respectively.

Aops community on internet has made efforts on it here Providing hypothesis... Proving a special case ... Hope we get to see a well defined constructive proof which even generalizes the Special case within itself.

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  • $\begingroup$ I can't find it now, but I recall a very similar question on this site just a few days ago. That post might have some useful info. for you in any comments and/or answers. $\endgroup$ – John Omielan Aug 23 at 5:56
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    $\begingroup$ Actually It was mine post which I deleted few days ago, aslike it really needed to framed in MSE culture. $\endgroup$ – ALPHATRION Aug 23 at 6:13
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    $\begingroup$ As it is your question doesn't make sense because almost every such problem about prime numbers is unprovable at least with the tools we have access to. Thus to ask for a (dis)proof you need to find some approach / article letting us think that problem is the exception and is provable. $\endgroup$ – reuns Aug 23 at 6:19
  • $\begingroup$ Closest I got was examining it mod 9 where there are only 4 squares ( 0,1,4,7); and 3 cubes (0,1,8). which can force you onto which arithmetic progressions of primes ? $\endgroup$ – Roddy MacPhee Aug 23 at 16:53
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    $\begingroup$ Code and it's output. language : PARI/GP $\endgroup$ – Roddy MacPhee Aug 26 at 12:04

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