Exponents of Known Mersenne Primes (Numerical Observations)

[Dobri]

Assuming a linear tendency for the approximation of Log[Abs[x]] and Log[Abs[y]] of the coordinates x and y of the prime exponent of the next Mersenne prime (if any) in Ulam clockwise square spiral, a lower-limit approximation could be made by setting the vertical intercept of the linear fit to zero.

Then the interval (5084, 5974) is obtained as a linear approximation of at least one of the next Abs(x) or Abs(y).

SetDirectory[NotebookDirectory[]];fname = NotebookDirectory[] <> "MersenneUlamX&YSortLogAbsLinearFit.jpg";
MpSx = {1, 0, -1, 0, 1, -2, -2, 2, -1, 5, -5, 6, 4, -5, -1, 18, -1, -3, 33, -32, -49, 9, -30, 71, 74, -76, 82, 46, -166, 182, 99, -374, 464, 536, -591, 863, -869, -967, 148, -2291, -374, 1717, -1018, -2854, 1501, -3265, -3283, -3804, -4307, -4394, -2733};
MpSy = {-1, -1, 0, 1, -2, -1, 1, -3, -4, -4, 2, -1, 11, 12, -18, 23, -24, 28, 4, 33, 36, -50, -53, 14, -19, 29, 105, -147, 113, -99, 232, -435, 359, -561, 554, 266, -136, 1320, -1835, -1004, 2451, -2548, -2757, -1461, -3048, -364, -3030, -307, 1978, -421, -4544};
nMp = Length[MpSx]; ic = 0; While[ic < nMp, ic++; MpSx[[ic]] = Log[Abs[MpSx[[ic]]]]; MpSy[[ic]] = Log[Abs[MpSy[[ic]]]];];
MpSx = Sort[MpSx]; MpSy = Sort[MpSy]; MpSxt = ConstantArray[0, {nMp - 8, 2}]; MpSyt = ConstantArray[0, {nMp - 8, 2}];
ic = 8; While[ic < nMp, ic++; MpSxt[[ic - 8, 1]] = ic; MpSyt[[ic - 8, 1]] = ic; MpSxt[[ic - 8, 2]] = MpSx[[ic]]; MpSyt[[ic - 8, 2]] = MpSy[[ic]];];
MpSxtZfit = FindFit[MpSxt, a*x, {a}, x]; xc = Exp[a*52 /. MpSxtZfit[[1]]];
MpSytZfit = FindFit[MpSyt, a*x, {a}, x]; yc = Exp[a*52 /. MpSytZfit[[1]]];
Print["(", Floor[yc], ",", Ceiling[xc], ")"];
ic = 8; While[ic < nMp, ic++; MpSxt[[ic - 8, 2]] = ic*a /. MpSxtZfit[[1]]; MpSyt[[ic - 8, 2]] = ic*a /. MpSytZfit[[1]];]
Show[ListLinePlot[{MpSx, MpSy, MpSxt, MpSyt}, PlotRange -> All, Frame -> True, PlotLegends -> {"x", "y", "x (linear fit)", "y (linear fit)"}], Frame -> True]
Export[fname, Show[ListLinePlot[{MpSx, MpSy, MpSxt, MpSyt}, PlotRange -> All, Frame -> True, PlotLegends -> {"x", "y", "x (linear fit)", "y (linear fit)"}], Frame -> True]]

[Dobri]

Assuming a linear tendency for the approximation of Log[Abs[x-y]] of the coordinates x and y of the prime exponent of the next Mersenne prime M52 (if any) in Ulam clockwise square spiral, a lower-limit approximation could be made by setting the vertical intercept of the linear fit to zero (which intuitively is logical since the center of the spiral is the origin of spiral spin).

Then Abs[x-y] ≈ 8631 for M52 (if any) is obtained as a linear approximation of Abs[x-y] (see the attached image).

From post #67, https://mersenneforum.org/showpost.p...8&postcount=67, Abs[x] ≈ 5973 (obtained with Round[ ] instead of Ceiling[ ]) and Abs[y] ≈ 5084.
Therefore, four combinations for x and y are obtained:
x ≈ 5973 and y ≈ -2658;
x ≈ -5973 and y ≈ 2658;
y ≈ 5084 and x ≈ -3547; and
y ≈ -5084 and x ≈ 3547.
To select approximation intervals around the linear fit values, the standard deviations of the actual data around the linear fits can be used.

[Dobri]

The prime with coordinates closest to x ≈ 5973 and y ≈ -2658 is M142691651 (Untested) with x = 5973 and y = -2654.

The prime with coordinates closest to x ≈ -5973 and y ≈ 2658 is M142715557 (Factored) with x = -5973 and y = 2668.

Attached for both cases are the lists of nearby primes and their x-y coordinates on the Ulam clockwise square spiral in the narrow intervals x ± ∆x and y ± ∆y, ∆x = ∆y = 100.

[Dobri]

Let's consider the intersection between LCS-generated sets and Ulam-clockwise-square-spiral-(UCS)-generated sets.

The intersections between the 28-bit LCS set in post #40, https://mersenneforum.org/showpost.p...1&postcount=40, and
the two distinct 28-bit UCS sets in post #69, https://mersenneforum.org/showpost.p...9&postcount=69,

give the following intersection subsets in the middle of their respective two-set Venn diagrams:

M144609329

for the UCS set with a central point x ≈ 5973 and y ≈ -2658;

and

M139862809,
M141666137,
M141666163,
M141666199, and
M144344789

for the UCS set with a central point x ≈ -5973 and y ≈ 2658.

Note: The same can be done also for the UCS sets with central points: y ≈ 5084 and x ≈ -3547; and y ≈ -5084 and x ≈ 3547. However, their 27-bit range from 99,xxx,xxx to 107,xxx,xxx is below the current wavefront for first-time tests.

[Dobri]

For completeness, here is the code used to obtain the linear fit of Log[Abs[x-y]] in post #68, https://mersenneforum.org/showpost.p...7&postcount=68.

(* Wolfram code *)
SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "MersenneUlam(X-Y)SortLogAbs.jpg";
MpSx = {1, 0, -1, 0, 1, -2, -2, 2, -1, 5, -5, 6, 4, -5, -1, 18, -1, -3, 33, -32, -49, 9, -30, 71, 74, -76, 82, 46, -166, 182, 99, -374, 464, 536, -591, 863, -869, -967, 148, -2291, -374, 1717, -1018, -2854, 1501, -3265, -3283, -3804, -4307, -4394, -2733};
MpSy = {-1, -1, 0, 1, -2, -1, 1, -3, -4, -4, 2, -1, 11, 12, -18, 23, -24, 28, 4, 33, 36, -50, -53, 14, -19, 29, 105, -147, 113, -99, 232, -435, 359, -561, 554, 266, -136, 1320, -1835, -1004, 2451, -2548, -2757, -1461, -3048, -364, -3030, -307, 1978, -421, -4544};
nMp = Length[MpSx]; MpSxy = ConstantArray[0, nMp]; MpSxyt = ConstantArray[0, {nMp, 2}];
ic = 0; While[ic < nMp, ic++; MpSxy[[ic]] = Log[Abs[MpSx[[ic]] - MpSy[[ic]]]];]; MpSxy = Sort[MpSxy];
MpSxyZfit = FindFit[MpSxy, a*x, {a}, x]; xc = Exp[a*52 /. MpSxyZfit[[1]]]; Print["a*52 -> ", Round[xc]];
ic = 1; While[ic < nMp, ic++; MpSxyt[[ic, 1]] = ic; MpSxyt[[ic, 2]] = ic*a /. MpSxyZfit[[1]];];
Show[ListLinePlot[{MpSxy, MpSxyt}, PlotRange -> All, Frame -> True, PlotLegends -> {"Log[Abs[x-y]]", "Log[Abs[x-y]] (linear fit)"}], Frame -> True]
Export[fname, Show[ListLinePlot[{MpSxy, MpSxyt}, PlotRange -> All, Frame -> True, PlotLegends -> {"Log[Abs[x-y]]", "Log[Abs[x-y]] (linear fit)"}], Frame -> True]]

[Dobri]

The 28-bit LCS-generated set in post #40 is obtained without the use of bit reversal and cyclic folding operations.

Using a larger LCS set involving said operations and also a larger UCS set (by selecting larger x ± ∆x and y ± ∆y intervals around the central points of the linear fit of Log[Abs[x-y]] in post #68) would give a richer combinatorial variety of prime exponents.

However, the intersection subset between the LCS-generated and UCS-generated sets would remain small, containing only several hundred 28-bit prime exponents to be tested for the first time.

[Dobri]

Let ∆x = ∆y = 1500. Then the intersection subset between the LCS-generated and UCS-generated sets around each central point contains several hundred 28-bit prime exponents to be tested for the first time (see the attached files).

The representation of the prime exponents in two (or more) dimensions (of which the x-y UCS representation is just an example) deserves a further examination to eventually obtain additional constraints and further reduce the number of prospective prime exponents within the framework described in previous posts.

[Dobri]

The linear tendency of the sorted array of Log[Abs[x-y]] of the x-y coordinates of the prime exponents of known Mersenne primes is preserved also on a modified Ulam clockwise square spiral of 𝜋(p) corresponding only to the prime numbers and omitting all composite integers.
At the center of said modified spiral is 𝜋(1) = 0.
The linear fit is shifted to the right from the origin (0, 0) though (see the attached image).

(* Wolfram code *)
SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "MersenneUlamPi(X-Y)SortLogAbs.jpg";
MpSx = {1, 1, 0, -1, -1, 0, 1, 2, -2, 2, 3, 2, -3, -4, -7, -9, -9, 11, -12, -10, -11, 18, 7, 24, 25, 16, -34, 34, -51, 56, 41, -123, 131, 156, 164, -144, 234, 345, 115, 577, -563, -2, 359, 357, 425, 532, -296, -150, -248, 1063, 799};
MpSy = {0, -1, -1, -1, 1, 1, 1, -1, 0, 2, -1, -3, -5, 5, 4, -5, 6, -4, -5, 12, 17, 16, 18, -22, -10, 25, -34, -46, 34, 45, 69, 106, -50, -141, 101, -232, 83, 157, 468, 259, 614, -637, -686, 708, 753, -804, -808, -928, -1043, -715, -1097};
nMp = Length[MpSx]; MpSxy = ConstantArray[0, nMp]; xt = 9; MpSxyt = ConstantArray[0, {nMp - xt, 2}];
ic = 0; While[ic < nMp, ic++; MpSxy[[ic]] = Log[Abs[MpSx[[ic]] - MpSy[[ic]]]];]; MpSxy = Sort[MpSxy];
ic = 0; While[ic < nMp - xt, ic++; MpSxyt[[ic, 1]] = ic + xt; MpSxyt[[ic, 2]] = MpSxy[[ic + xt]];];
MpSxyFit = FindFit[MpSxyt, a*x + b, {a, b}, x]; yi = b /. MpSxyFit[[2]];
ic = 0; While[ic < nMp - xt, ic++; MpSxyt[[ic, 2]] = (ic + xt)*a /. MpSxyFit[[1]]; MpSxyt[[ic, 2]] = MpSxyt[[ic, 2]] + yi];
xyc = (nMp - xt + 1)*a /. MpSxyFit[[1]]; xyc = xyc + yi; xyc = Exp[xyc]; Print["a*52 -> ", Round[xc]];
Show[ListLinePlot[{MpSxy, MpSxyt}, PlotRange -> All, Frame -> True, PlotLabel -> "A graph based on Ulam Clockwise Square Spiral of" PrimePi[p], PlotLegends -> {"Log[Abs[x-y]]", "Log[Abs[x-y]] (linear fit)"}], Frame -> True]
Export[fname, Show[ListLinePlot[{MpSxy, MpSxyt}, PlotRange -> All, Frame -> True, PlotLabel -> "A graph based on Ulam Clockwise Square Spiral of" PrimePi[p], PlotLegends -> {"Log[Abs[x-y]]", "Log[Abs[x-y]] (linear fit)"}], Frame -> True]]

[Dobri]

The assumed linear tendencies of sorted arrays of ln(|x|), ln(|y|), and ln(|x-y|) correspond to an equation for which there is no known analytical solution:

sgn(x) e^{linear term 1} + sgn(y) e^{linear term 2} = sgn(x-y) e^{linear term 3}, n = 1, 2, 3,..., 51, 52,...

Without loss of generality, the base can be changed from e to 2:

sgn(x) 2^an+b + sgn(y) 2^cn+d = sgn(x-y) 2^fn+g, n = 1, 2, 3,..., 51, 52,...

[Dobri]

Originally Posted by Dobri

...
sgn(x) e^{linear term 1} + sgn(y) e^{linear term 2} = sgn(x-y) e^{linear term 3}, n = 1, 2, 3,..., 51, 52,...
...
sgn(x) 2^an+b + sgn(y) 2^cn+d = sgn(x-y) 2^fn+g, n = 1, 2, 3,..., 51, 52,...

This is a correction of an apparent typo in the previous post. The correction replaces a positive sign with a negative sign as follows:
sgn(x) e^{linear term 1} - sgn(y) e^{linear term 2} = sgn(x-y) e^{linear term 3}, n = 1, 2, 3,..., 51, 52,...
sgn(x) 2^an+b - sgn(y) 2^cn+d = sgn(x-y) 2^fn+g, n = 1, 2, 3,..., 51, 52,...

Food for thought: What IF the deviations of the actual x-y coordinates of the prime exponents of known Mersenne primes from the linear fits are due to a replacement of the real coefficients a, b, c, d, f and g with integer coefficients A, B, C, D, F and G thus forming the closest Diophantine expressions to the linear fits (or not)?

[Dobri]

In addition to the linear tendency of the sorted array of ln(|x-y|) of the x-y coordinates of the prime exponents of known Mersenne primes on a modified Ulam clockwise square spiral of 𝜋(p) in post #74 (https://mersenneforum.org/showpost.p...8&postcount=74), a second linear tendency of the sorted array of ln(min(|x|,|y|)) is also observed (see the attached image) while the sorted array of ln(max(|x|,|y|)) has a rather nonlinear tendency (green curve).

From the linear fits of the two linear tendencies, the approximations |x-y| ≈ 2497 and min(|x|,|y|) ≈ 1070 for n = 52 are obtained.
This gives four local points for 28-bit prime exponents, M144312713 (factored), M144325877 (factored), M144419089 (factored), and M144431951 (factored).
The result is close to the local points M142691651 (now factored) and M142715557 (factored) in post #69 (obtained on the standard Ulam clockwise square spiral, https://mersenneforum.org/showpost.p...9&postcount=69).