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A Diophantine equation has integer coefficients and integer exponents, and integer solutions are studied. The number of terms is finite.
A1 X1N1 + A2 X2N2 + A3 X3N3 + ... = 0
A transcendental Diophantine equation is any equation for which integer solutions are studied.
Arrows show corollaries.
Equation Theorems and conjectures A * B = C Prime numbers. Twin prime conjecture, Prime gaps, Prime clusters A + B = C Goldbach conjecture Mason-Strothers conjecture -> ABC conjecture Baker conjecture -> ABC conjecture -> Mordell theorem -> Fermat theorem -> General Pilai conjecture AX + BY = C Generalized Eliot-Halberstam conjecture -> Eliot-Halberstam conjecture Polignac conjecture -> Twin prime conjecture Goldbach conjecture X1N + X2N + ... + XTN = 0 Ekl conjecture, Euler's sum-of-powers counterexamples Langlands reciprocity conjecture -> Modularity theorem -> Fermat theorem -> Birch & Swinnerton-Dyer conjecture Geometric Langlands conjecture AX + BY = CZ ABC -> Fermat-Catalan conj -> Beal conj -> Generalized Fermat theorem -> Fermat theorem Prime gaps Generalized Riemann Hypothesis -> Riemann hypothesis Cramer conjecture -> Brocard conjecture -> Legende conjecture Oppermann conjecture -> Brocard conjecture -> Legende conjecture Oppermann conjecture -> General Andrica conjecture -> Andrica conjecture Firoozbakht conjecture -> Cramer upper bound FGKMT lower bound Bateman-Horn conjecture -> First Hardy-Littlewood conjecture Gilbreath conjecture X1 + X2 + ... + XN = 0 Strong n conjecture -> n conjecture -> ABC conjecture M/N = X-1 + Y-1 + Z-1 Generalized Erdos-Straus conjecture -> Erdos-Straus conjecture for M=4 A XM - B YN = C Pillai conjecture -> Hall conjecture 2N + 1 = P + 2Q Lamoine conjecture -> Goldbach weak conjecture 2N + 1 = P + M(M+1) Sun conjecture X2 + Y2 <= R2 Generalized Gauss circle problem -> Gauss circle problem N! + 1 = M2 Brocard problem XY - YX ABC conjecture -> Weak Hall conjecture Plane set, rational dist Bombieri-Lang conjecture -> Erdos-Ulam false ABC conjecture -> Erdos-Ulam false Schinzel hypothesis H -> Dickson conjecture -> Twin prime conjecture -> Sophie Germain conjecture
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The maximal gap function Gap(X) is the size of the biggest prime gap up to prime X.
A "record" gap at X is such that there are no known gaps of larger magnitude at smaller X. It doesn't necessarily mean that they don't exist. In the chart, all known record gaps are shown.
Some primality tests are guaranteed to correctly distinguish between a prime and composite. Some tests are probabilistic. They have good accuracy but occassionally a composite passes the test. Such a composite is called a "pseudoprime".
In the chart, "guaranteed gaps" are record gaps with guaranteed primes. "Probable gaps" are record gaps with probable primes.
The average gap size at X scales as ln(X). The scaling for the maximal gap is poorly constrained, as is the gap distribution.
If gap size G is Gaussian distributed as e-(G/lnX)2, the maximal gap scales as Gap(X) ~ ln3/2X.
If gaps are exponentially distributed as e-G/lnX, the maximal gap scales as Gap(X) ~ ln2X. This is the Cramer conjecture.
Numerical evidence supports an exponential distribution but there is a computational limit for how far we can explore the tail. The limit hinges on sieve size and memory.
Gap(X)
Upper bound, proven, 1852 X Assumes a prime between P and 2P. Bertrand postulate
Upper bound, proven, 1930 X32999/33000 Hoheisel
Upper bound, proven, ? X249/250 Heilbronn
Upper bound, proven, 1936 X3/4 Cudakov
Upper bound, proven, 1937 X5/8 Ingram Implies a prime between successive cubes
Upper bound, proven, 1972 X7/12 Huxley
Upper bound, proven, 2001 X.525 Baker, Harman, & Pintz
Upper bound, conjecture, 1912 X1/2 Legendre conj. Assumes a prime between successive squares
Upper bound, conjecture, 1969 X1/2/ln(X) Grimm conjecture
Upper bound, conjecture, 1936 ln(X) ln(X) Cramer
Upper bound, conjecture, 1982 ln(x) ln(x) - ln(x) - 1 Firoozbakht Assumes PN1/N strictly decreasing, where PN is the Nth prime
Riemann hypothesis corollary 1936 X1/2 ln(X) Cramer
Lower bound, proven, 1931 ln(X) Westzynthius
Lower bound, proven, 1938 ln(X) lnln(X) lnlnlnln(X) / [lnlnln(X)]2 Rankin
Lower bound, proven, 2018 ln(X) lnln(X) lnlnlnln(X) / lnlnln(X) Ford-Green-Konyagin-Maynard-Tao
Define an Ekl polynomial Ekl(N,U,V).
X1N + X2N + ... + XUN = Y1N + Y2N + ... + YVN Q = U + V - N
The Ekl conjecture is that all integer solutions have Q>=0. The Fermat theorem follows.
Denote a solution with brarckets. For example, a solution for Ekl(2,1,2) is:
52 = 32 + 42 <--> [5] = [3 4]
Euler's sum of powers conjecture is that Q>=1 for integer solutions of Ekl(N,1,V). It was proven false with counterexamples for N=4 and N=5. No counterexamples exist for N>5. The smallest counterexample for N=4 and N=5 is
4224814 = 958004 + 2175194 + 4145604 1445 = 275 + 845 + 1105 + 1335
Solutions for Ekl(N,1,V) with Q=1 are known only for N = {2, 3, 4, 5, 7, 8}
The First Hardy-Littlewood conjecture (1923) is that the density of prime M-tuples at X is 1/lnM(X).
The prime number theorem is the case M=1 and was proven in 1896 by Hadamard & Poussin independently.
In 1915, Brun proved that the number of twin primes less than X is less than (2/3) X/lnM(X).
The Bateman-Horn conjecture (1962) generalizes the First Hardy-Littlewood conjecture.
The Polignac conjecture (1849) is that there are infinitely many prime M-tubles for all M.
t was proved for M >= 70,000,000 by Zhang in 1013, and for M >= 246 by the Polymath8 project in 2014.
If the Eliot-Halberstam conjecture is true, then the Polignac conjecture is true for M >= 12. If the generalized Eliot-Halberstam conjecture is true, then the Polignac conjecture is true for M >= 6.
Generalized Fermat equation:
AX + BY = CZ
Beal conjecture: All integer solutions with {X, Y, Z} > 2, {A, B, C} have a common prime factor.
The Fermat-Catalan conjecture is that there are only finitely many solutions with A, B, and C being positive integers with no common prime factor and X, Y, and Z being positive integers satisfying X-1 + Y-1 + Z-1 < 1
The Beal conjecture can be restated as "All Fermat-Catalan conjecture solutions use 2 as an exponent".
The ABC conjecture implies that there are at most finitely many counterexamples to the Beal conjecture.
Proofs for specific exponents:
Exponent
3 1770 Euler
4 1670 Fermat
5 1825 Legendre; Dirichlet
6 1802 Kausler
7 1869 Lame
10 1913 Kapferer
14 1832 Dirichlet
<270 1823 Germain
<2522 1954 Vandiver Computer proof
<125000 1978 Wagstaff Computer proof
<4000000 1993 Computer proof
Even 1977 Terjanian
Regular prime Kummer Conjecture: 61% of primes are regular
All 1995 Wiles
1955 Modularity conjecture. Taniyama & Shimura
1983 Falting theorem
1984 Frey theorem
1986 Ribet theorem
2001 Modularity theorem
The Catalan conjecture was posed in 1842: The only solution to AX - BY = 1 is 32 - 23 = 1. It was proved in 2002 by Mihailescu.
To find zeros of the Riemann Zeta function, use the Hardy Z function, which is real-valued function along the critical line.
Search for sign changes. Use root-finding (Newton, Brent, secant, etc.). Count how many zeros were found. Independently count how many zeros should exist using the Riemann-von Mangoldt formula. If the counts agree, every zero up to that height lies on the critical line. This is the method used in essentially every large-scale verification.
The Odlyzko-Schonhage formula evaluates all zeros in an interval collectively, at a cost of ln(X) per zero.
The density of zeros is ln(X)
For evaluating the Hardy function at a single point, the Riemann-Siegel formula scales as X½.
Calculations use interval arithmetic, ball arithmetic, and arbitrary precision libraries. Each computed value is an interval guaranteed to contain the true value. Packages like Arb (now integrated into FLINT) make this practical.
The Riemann hypothesis is equivalent to all Li coefficients being positive, although this test is more expensive than evaluating zeros.
The scaling for RMS magnitude is H(X) = ln(X)
The scaling for moments of order m is [ln X]m2/4 Keating-Snaith random matrix prediction
To search for solutions to Euler polynomials Elk(N,1,N) up to height H, the naive computational scaling is N HN / (N-1)!.
Meet-in-the-middle technique improves the scaling to N H(N-1)/2.
Modular sieving can reduce the search space and reduce the constant in front of the scaling.
The generalized Andrica conjecture is PN+1Z - PNZ < 1 if Z < .670873.... where PN+1 is the Nth prime.
The Andrica conjecture is for the case Z=1/2.
There are at least 1 primes between successive squares. This implies that there are at least 2 primes between successive prime squares.
There are at least 4 primes between successive prime squares. The Brocard conjecture implies th Legendre conjecture.
Some tests are guaranteed to correctly distinguish between a prime and composite. Some tests are probabilistic. They have good accuracy, but occassionally a composite passes the test. Such a composite is called a "pseudoprime".
Scaling
Baillie-PSW Probable
Adleman-Pomerance-Rumely [ln X]ln ln ln X
Agrawal-Kayal-Saxena Guarantee [ln X]6 2002
Agrawal-Kayal-Saxena + Agarwal conj. Guarantee [ln X]3 2005
The Baillie–PSW primality test is a probabilistic or possibly deterministic primality testing algorithm that determines whether a number is composite or is a probable prime. It is named after Robert Baillie, Carl Pomerance, John Selfridge, and Samuel Wagstaff.
The Baillie–PSW test is a combination of a strong Fermat probable prime test to base 2 and a standard or strong Lucas probable prime test. The Fermat and Lucas test each have their own list of pseudoprimes, that is, composite numbers that pass the test. There is no known overlap between these lists, and there is even evidence that the numbers tend to be of different kind, in fact even with standard and not strong Lucas test there is no known overlap.
Computers show that there no Baille-PSW pseudoprimes below 1.9e18.
The conjecture is that zeros of the Riemann zeta function have a correlation function of
1 - [(sin(π X) / (π X)]2
The conjecture has big numerical support.
Let X have B bits. The cost of integer multiplication is M(B) = B2. FFT techniques can sometimes reduce this to M(B) = B log2B.
Digits X Reach scaling
Reach, Riemann zeros 12 4e11 M(B) X ln(X)
Reach, Euler cuboid 13 e12 M(B) X3
Reach, Erdos-Straus conjecture 18 e17 M(B) X
Reach, Goldbach conjecture 19 4e18 M(B) X ln ln X
Reach, ABC conjecture 20 e19
Reach, Firoozbakht conjecture 19 4e18
Reach, prime number sieve 20 e19 M(B) X ln ln(X) Memory is X½
Reach, prime gap, maximal 20 e19
Reach, prime gap, proven 7 1113106
Reach, prime gap, probable 8 16045848
Reach, Collatz conjecture 20 e20 X
Reach, Ekl(6,1,5) 730000 X5/2
Reach, Ekl(6,1,6) X3
Reach, Ekl(7,1,5) X5/2
Reach, Ekl(8,1,7)
Reach, Ekl(9,1,8)
Reach, Ekl(9,1,9)
Reach, Ekl(7,3,4)
Reach, Ekl(9,4,5)
Reach, Ekl(10,5,5)
Reach, Li coefficient
Reach, Catalan-Beal conj.
Reach, Aliquot sequence
Reach, Agrawal conjecture 18 e17
Reach, Andrica conjecture 20 2e19
Reach, Hall conjecture 29 6e28
Reach, Lamoine conjecture 11 1e10
Reach, Gilbreath conjecture 20 1.5e15
Reach, Brocard problem 16 1e16 Known solutions are (4,5), (5,11), (7,71)
Biggest Mersenne prime 41000000 Big
Biggest probable prime 8200000 Big M(B) ln(B) Miller-Rabin PRP test
Biggest twin prime 388342 Big
Biggest triplet prime 20008 Big
Biggest Riemann zero 35 8e34 M(B) X½
Skewes point, proven 317 1.49e316
Skewes point, min 175 1.5e174
Skewes point, prime pair 7 1369391 Wolf 2011
Skewes point, prime triple 6 337867 Toth 2019
Skewes point, prime 4-tuple 7 1172531 Toth 2019
Skewes point, prime 5-tuple 8 21432401 Toth 2019
Skewes point, prime 6-tuple 12 Toth 2019 251331775687
Skewes point, prime 7-tuple 13 Pfoertner 2020 7572964186421
Skewes point, prime 8-tuple 16 Pfoertner & Luhn 2020 1203255673037261
Mertens lower bound 17 e16
Mertens upper bound 8.5e18 Big
Polya conj min counterexample 9 906150257 Tanaka 1980
Archimedes sand reckoner # 63 Big
Googol 100 Big
Googolplex 10100 Big
Integer, 4 byte 10 2e9
Integer, 8 byte 19 9e18
Integer, 16 byte 39 2e38
Integer, 32 byte 77 6e76
Float, 4 byte 39 3e38
Float, 8 byte 309 2e308
Float, 16 byte 4933 1e4932
1 = 32 - 23 24 = 210- 103 13 = 28 - 35 3 = 27 - 53 2 = 33 - 52 104 = 36 - 54 1 = 23 - 7 4 = 53 - 112 7 = 27 - 112 7153 = 312- 219 1 = 2*5 - 32 5 = 53 - 3*23 1 = 24 - 3*5 2 = 25 - 2*3*5 5 = 7*15 - 102 1.75 ~ (3/2)12 - 27 Pythagorean comma. Relates musical octaves to fifths. 1.024= 27/53 = 210/103 7/5 ~ 2½ Pi ~ 22/7 e^P ~ 20 e^e ~ 15.15
Clusters:
7, 8, 9, 10 25, 27, 30, 32
The tables gives minimum value of Q known from numerical examples, along with the minimal example. Solutions for Ekl(N,1,2) are ruled out by the Fermat theorem.
N U V Q 2 1 2 1 [5] = [3 4] Pythagorean triples Antiquity 3 1 3 1 [6] = [3 4 5] Antiquity 4 1 3 0 [422481]= [95800 217519 414560] Euler counterexample 1998 Elkies 5 1 4 0 [144] = [27 84 110 133] Euler counterexample 1967 Lander et al. 6 1 7 2 [1141] = [76 234 402 474 702 894 1077] 1967 Lander et al. 7 1 7 1 [568] = [525 439 430 413 266 258 127] 1999 Dodrill 8 1 8 1 [1409] = [1324 1190 1088 748 524 478 223 90] Chase; Meyrignac Only solution known 9 1 10 2 [917] = [851 822 668 625 574 542 475 179 99 42] 10 1 13 4 [228] = [210 204 187 179 128 122 85 73 59 57 49 13 6] Chase 2 2 2 2 [2 11] = [5 10] Antiquity 3 2 2 1 [1 12] = [9 10] Antiquity 4 2 2 0 [59 158] = [133 134] 1772 Euler 5 2 3 0 [141325 2205] = [140685 62375 50275] 1997 Scher & Seidl 6 2 5 1 [770 1117] = [84 212 602 861 1092] 1999 Brisse 7 2 6 1 [125 24] = [121 94 83 61 57 27] Meyrignac 8 2 7 1 [1303 1127]= [1334 976 648 623 516 401 272] Chase; Meyrignac Only solution known 9 2 9 2 [137 69] = [121 116 116 115 89 52 28 26 14 9] 2002 Wroblewski 10 2 12 4 [112 99] = [109 103 89 79 72 59 59 52 20 15 5 5] 2000 Pliousnine 2000, 2000 Kuosa 5 3 3 1 [3 54 62] = [24 28 67] 1967 Lander et al. 6 3 3 0 [3 19 22] = [10 15 23] 1934 Subba Rao 7 3 5 1 [96 41 17] = [87 77 77 68 56] 8 3 5 0 [966 539 81]=[954 725 481 310 158] 2003 Chase, Meyrignac, Resta & Meyrignac 9 3 9 3 [38 38 3] = [41 23 20 20 18 13 13 12 9] 1998 Ekl 10 3 11 4 Solutions known 2002 Wroblewski 6 4 4 2 [2 2 9 9] = [3 5 6 10] 1934 Rao 7 4 4 1 [149 123 14 10] = [146 129 90 15] 1996 Ekl 8 4 4 0 [3113 2012 1953 861] = [2823 2767 2557 1128] 2006 Kuosa 9 4 6 1 [90 64 35 35] = [86 80 62 43 27 16] 10 4 9 3 Solutions known 2002 Wroblewski 9 5 5 1 [192 101 91 30 26] = [180 175 116 17 12] 1997 Ekl 10 5 16 11 Solutions known 10 6 6 2 [95 71 32 28 25 16] = [92 85 34 34 23 5] 2002 Kuosa
In the table, the first column is N and the top row is U. The interior is the minimum known Q for integer solutions of Ekl(N,U,V) with U <= V.
U -> 1 2 3 4 5 6
N
2 1 2
3 1 1
4 0 0
5 0 0 1
6 2 1 0 2
7 1 1 1 1
8 1 1 0 0
9 2 2 3? 1 1
10 4 4 4 3 7? 2
Tetration can represent numbers from 0 to eeeeee on the same scale. It goes beyond a logarithmic scale.
Define "tetration":
Tet(0) = 1
Tet(1) = e
Tet(2) = ee
Tet(3) = eee
Tet(Q+1) = eTet(Q)
Define a continuous function Tet(Q)=X.
It has an analytic form that is complicated to express.
We use a piecewise interpolation with unit intervals.
Supercomputers today have a speed of 1018 Flop/second.
A computer powered by a blue giant star has a speed of order 1048
Flop/second. A computer powered by a galaxy cluster has a speed of order 1056 Flop/second.
Euler polynomials
https://www.tandfonline.com/doi/full/10.1080/0025570X.2025.2481010#abstract Prime gaps
Acknowledgments: The author
thanks ChatGPT (OpenAI) for discussions that helped organize ideas, clarify
terminology, and explore possible frameworks for presenting computational reach
in mathematics.
-1 < Q < 0 0 < X < 1 X = Q+1 Q = -1 + X
0 < Q < 1 1 < X < e X = eQ Q = 0 + ln(X)
1 < Q < 2 e < X < ee X = ee(Q-1) Q = 1 + ln ln(X)
2 < Q < 3 ee< X < eee X = eee(Q-2) Q = 2 + ln ln ln(X)
etc.
Speed, computer in 2026 = e18 Flop/second
Speed, blue giant computer = e48 Flop/second
Speed, galaxy cluster computer= e56 Flop/second
Power, blue giant star = e32 Watt
Power, large galaxy = e36 Watt
Power, galaxy cluster = e40 Watt
Energy per Flop, 2026 = 1e-12 Joule/Flop
Energy per flop, far future = 1e-16 Joule/Flop
Lifetime, calculation of today= 3e7 second = 1 year
Lifetime, blue giant star = 3e13 second = 10 million years
Lifetime, galaxy cluster = 3e17 second = 10 billion years
Computer Flop total, 2026 = 3e25 Flop
Computer Flop total, blue giant = 3e61 Flop
Computer Flop total, galaxy cluster= 3e73 Flop
Years
Summer and winter switch 13000
Supervolcano timescale, 1 km3 17000
Supervolcano timescale, 1000 km3 100000
Supervolcano timescale, 3000 km3 500000
Asteroid timescale, 1 km 1 mil
The African rift becomes an ocean 10 mil
Lifetime of a blue giant star 10 mil
Atlantic ocean volcano ring forms 20 mil
San Andreas fault moves LA and SF together 50 mil
Asteroid timescale, 10 km 100 mil
Atlantic changes from expand to contract 125 mil
Sun orbit time around Galaxy 240 mil
Supercontinent forms 250 mil
Pacific Ocean stops expanding 350 mil
Plate tectonics stop 650 mil
Oceans evaporate. Moon outspiral stops 1100 mil
Mars becomes warm enough for liquid water 1550 mil
Sun nova 7590 mil
Milky Way and Andromeda merge 10 bil
Big Rip, minimum time 22 bil
Galaxies outside the LG are gone 125 bil
Local Group galaxies merge 300 bil
Universe gravitationally segments 325 bil
Red dwarf lifetime 15 tril
Star formation ends 100 tril
Proton decay e43
Black hole evap, 1 solar mass e67
Biggest supermassive black holes evaporate e109
Vacuum collapse e161
Inflationary event from tunneling 10^10^10^56
Euler polynomials. Mathworld
Prime gap list
Euler polynomials
Big primes. t5.org
Encyclopedia of integer sequences
Prime gaps
Prime gaps
Ekl, R. L. "Equal Sums of Four Seventh Powers." Math. Comput. 65, 1755-1756, 1996.
Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.
Elkies, N. "On A^4+B^4+C^4=D^4." Math. Comput. 51, 828-838, 1988.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 195, 1998.
Lander, L. J. and Parkin, T. R. "A Counterexample to Euler's Sum of Powers Conjecture." Math. Comput. 21, 101-103, 1967.
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446-459, 1967
Letac, A. Gazetta Mathematica 48, 68-69, 1942.
Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." http://euler.free.fr.
Moessner, A. "Einige Numerische Identitaten." Proc. Indian Acad. Sci. Sect. A 10, 296-306, 1939.
Subba Rao, K. "On Sums of Sixth Powers." J. London Math. Soc. 9, 172-173, 1934.
Wiles 1994
E. Brisse 1999
Resta 1999
Resta and Meyrignac 2003
1934 Subba Rao
M. Dodrill 1999, PowerSum
Erik Westzynthius
Largest known zero of the Riemann zeta function https://mathoverflow.net/questions/264052/largest-known-zero-of-the-riemann-zeta-function
© Jason Maron, all rights reserved.
Data from Wikipedia unless otherwise specified.