Somnath Maiti | Searcharxiv

arXiv2026

Multiple and Complete New Important Conjectures on Perfect Cuboid and Euler Brick

Nobody has discovered any perfect cuboid and there is no formula to deliver all possible Euler bricks. During investigations of famous open problems regarding the perfect cuboid and Euler brick; I have found new important conjectures on Pythagorean triples and biquadratic Diophantine equations [4] which are reduced $\&$ complete form for perfect cuboid and Euler brick problems. The details of the conjectures have been provided in Sections 2-3. If any perfect cuboid exists, it will be only among the solutions of six conjectures and all the Euler bricks are only among the solutions of next three conjectures [4]. For example, if any odd $n\in \mathbb{N}$ satisfy $n=e^2-f^2=g^2-h^2=k^2-l^2$ and $e^2f^2=g^2h^2+k^2l^2$; then we can discover a perfect cuboid of type 1 as $\{e^2-f^2,2gh,2kl,g^2+h^2,k^2+l^2,2ef,e^2+f^2\}$ having $(e^2-f^2,2gh,2kl)$ as its edges; $(g^2+h^2,k^2+l^2,2ef)$ as its face diagonals and $e^2+f^2$ as its body diagonal where $e,f,g,h,k,l~(>1)\in \mathbb{N}$. Equivalently, biquadratic Diophantine equation conjectures have been introduced for these perfect cuboid conjectures. For the benefit of readers, along with the original contribution for new important conjectures on perfect cuboid and Euler brick problems; brief review related to Pythagorean Triple, perfect cuboid and Euler brick problems as well as on Diophantine Equation and Biquadratic Diophantine Equation; studied in the past by previous researchers, have been discussed in the paper.

math.GM↗

arXiv2026

Results of Brocard-Ramanujan problem on diophantine equation $n!+1=m^2$

The Brocard-Ramanujan problem pertaining to the diophantine equation $n!+1=m^2$, a famously unsolved problem, deals with finding the integer solutions to the equation. Nobody has discovered any new solution of the problem beyond $n=4,~5$ and $7$ although many of us have tried it. Bruce Berndt and William Galway \cite{Berndt} had not found any new solution in 2000 by extensive computer search for a solution with $n$ up to $10^9$. The purpose of this study is to show that the solutions should satisfy some necessary and/or sufficient conditions. If $\sqrt{n!}=k+ε,~n>1,~0<ε<1$; then it has solution if and only if $n!=k(k+2)$ and $ε,~k$ are strictly monotonic increasing. It has only finitely many solutions which is not based on any conjecture or previous research on the Brocard-Ramanujan problem. For the new solution of Brocard-Ramanujan problem ($n\ge 10^5$), the value of $ε$ should be more than $0.999 \cdots 905915$ (digit 0 is coming after 228287 numbers of 9 digit, which takes more than 66 pages in (LibreOffice Writer) indicating almost impossibility of new solution. If we consider $n\geq 10^9$, I am unable to calculate the said numbers of 9 digit in the value of $ε$ in my personal laptop (with 8GB Ram) using MATHEMATICA 8. Finally, it has been claimed to discover that the problem has no further solution.

math.NT↗