To used the graphical method from (post "Beal And Integer Quadrant Arcs" dated 31 Dec 2022), we plot \(A^x=n_{j}\), \(B^y=1\) and \(C^z=n_{j}+1\). Any irreducible factor wholly divides \(n_{j}\) and is a prime factor of \(n_{j}\).
At first it seem, this graphical method to find factors that disallows other factors when a prime factor is plotted, would discontinue Grimm series of \(n_j\) immediately.
Arc \(n_j+1\) seem to be due to a factor of \(n_{j+1}+1\). However, \(n_{j+1}+1\) is not on the same plot as \(n_j+1\). A prime factor of \(n_{j+1}\) will exclude this line, OD`. OD and OD` is not coincidental.
However, for the integer \(n_{j+k}=4n_{j}+3\) with arc \(4(4n_{j}+3+1)=16(n_{j}+1)\), the arc \(4(n_{j}+1)\) suggest a factor \(4\).
What? Ignore this line when doing arc \(4n_{j}+3\)?Happy New Year 2023.
No, the line OD is on a integer point, markings made by dividing the quadrant arc by \((n+1)\); the arc is part of a circle with circumference \(4(n+1)\). This point and the origin give OD. All quadrant arcs draw with integer length that intersects line OD on a integer marking contributes a factor to \(n\). With OD line due to the prime factor, ono other lines are possible.
Any factor set up a series of marking on its arc that excludes all intersections with OD all other arcs not from factors, once OD is drawn.
There is only one unique OD.
Any arc of integer length, does not have markings that intersect OD drawn, except arcs from factors.
Maybe...
