Diophantine Equation - Revision history

HelpMe: update

2026-10-05T09:18:27Z

update

← Older revision		Revision as of 09:18, 5 October 2026
Line 111:		Line 111:

	== Cryptids ==		== Cryptids ==
	As of ~~July~~ 2026, Bogdan Grechuk lists 4 equations of size 32 whose integer solvability is currently unknown, the smallest size with unsolved equations.<ref>https://arxiv.org/pdf/2404.08518 A systematic approach to Diophantine equations: open problems, Bogdan Grechuk</ref>		As of October 2026, Bogdan Grechuk lists 3 equations of size 34 whose integer solvability is currently unknown, the smallest size with unsolved equations.<ref>https://arxiv.org/pdf/2404.08518 A systematic approach to Diophantine equations: open problems, Bogdan Grechuk</ref>

			There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10 times the compute used to solve the case of k = 42, so it is not as futile as standard Cryptids. sheep suggests calling it an [[User:RobinCodes/Machines at the Edge\|energy vampire]].

	There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.

Tlonuqbar: fixing link

2026-09-30T00:30:53Z

fixing link

← Older revision		Revision as of 00:30, 30 September 2026
Line 111:		Line 111:

	== Cryptids ==		== Cryptids ==
	As of July 2026, Bogdan Grechuk lists 4 equations of size 32 whose integer solvability is currently unknown, the smallest size with unsolved equations.<ref>https://arxiv.org/~~abs~~/~~2108~~.~~08705~~</ref>		As of July 2026, Bogdan Grechuk lists 4 equations of size 32 whose integer solvability is currently unknown, the smallest size with unsolved equations.<ref>https://arxiv.org/pdf/2404.08518 A systematic approach to Diophantine equations: open problems, Bogdan Grechuk</ref>

	There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.		There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.

Tlonuqbar: /* Cryptids */

2026-08-27T09:11:35Z

Cryptids

← Older revision		Revision as of 09:11, 27 August 2026
Line 111:		Line 111:

	== Cryptids ==		== Cryptids ==
	As of July 2026, Bogdan Grechuk lists 4 equations of size 32 whose solvability ~~in the integers~~ is currently unknown.<ref>https://arxiv.org/abs/2108.08705</ref>		As of July 2026, Bogdan Grechuk lists 4 equations of size 32 whose integer solvability is currently unknown, the smallest size with unsolved equations.<ref>https://arxiv.org/abs/2108.08705</ref>

	There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.		There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.

Tlonuqbar: formatting

2026-08-27T09:08:22Z

formatting

← Older revision		Revision as of 09:08, 27 August 2026
Line 115:		Line 115:
	There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.		There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.


			== References ==
			<references/>
	== See Also ==		== See Also ==

	* [https://discord.com/channels/960643023006490684/1461768250575687897 Discord BBdio thread]		* [https://discord.com/channels/960643023006490684/1461768250575687897 Discord BBdio thread]

	[[Category:Functions]]		[[Category:Functions]]

Tlonuqbar: lower bound on unsolved equations

2026-08-27T09:02:30Z

lower bound on unsolved equations

← Older revision		Revision as of 09:02, 27 August 2026
Line 111:		Line 111:

	== Cryptids ==		== Cryptids ==
			As of July 2026, Bogdan Grechuk lists 4 equations of size 32 whose solvability in the integers is currently unknown.<ref>https://arxiv.org/abs/2108.08705</ref>

	There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.		There are several open problems in the [[wikipedia:Sums_of_three_cubes\|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.

C7X: Ah you're right, thanks. Hopefully this one works, it seems to have been missed in Grechuk's preprint

2026-08-19T04:09:12Z

Ah you're right, thanks. Hopefully this one works, it seems to have been missed in Grechuk's preprint

← Older revision		Revision as of 04:09, 19 August 2026
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	\|		\|
	\|-		\|-
	\|48		\|33
	\|> 10<sup>10</sup>		\|> 10<sup>11</sup>
	\|<math>x^3 + y^3 + z^3 = 24</math>		\|<math>x^4 + y^2z + z^2 - z + 3 = 0</math>
	\|(~~-2901096694~~, -~~15550555555, 15584139827~~)*		\|(708313, 1099536, -267298981516)*
	\|https://~~oeis~~.org/~~A060467~~		\|https://arxiv.org/abs/2108.08705 v3 page 27
			\|-
			\|...
			\|
			\|
			\|
			\|
	\|-		\|-
	\|57		\|57

Sligocki: Undo revision 8336 by C7X (talk) I think this doesn't work. If you make it into a polynomial equation by multiplying across by denom, then (0, 0, 0) is a solution.

2026-08-19T04:02:34Z

Undo revision 8336 by C7X (talk) I think this doesn't work. If you make it into a polynomial equation by multiplying across by denom, then (0, 0, 0) is a solution.

← Older revision		Revision as of 04:02, 19 August 2026
Line 101:		Line 101:
	\|(12602123297335631, 80435758145817515, −80538738812075974)*		\|(12602123297335631, 80435758145817515, −80538738812075974)*
	\|https://oeis.org/A060467		\|https://oeis.org/A060467
	\|-
	~~\|352~~
	~~\|> 10<sup>80</sup>~~
	~~\|<math>\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} = 4</math>~~
	~~\|(<math> >4.373\times 10^{78}</math>, <math> >3.687\times 10^{79}</math>, <math> >1.544\times 10^{80}</math>)~~
	~~\|https://oeis.org/A369896~~
	\|}		\|}
	Note: For large champions (marked with * in Min Solution column) these are not known to be strictly the smallest magnitude solutions, instead they appear to be solutions with the minimal smallest value. If that is the case, then this still bounds BBdio since the score of these equations must be ≥ the minimal smallest variable value.		Note: For large champions (marked with * in Min Solution column) these are not known to be strictly the smallest magnitude solutions, instead they appear to be solutions with the minimal smallest value. If that is the case, then this still bounds BBdio since the score of these equations must be ≥ the minimal smallest variable value.

C7X: Please remove if this is not rigorous/well-sourced enough of an edit

2026-08-19T03:55:47Z

Please remove if this is not rigorous/well-sourced enough of an edit

← Older revision		Revision as of 03:55, 19 August 2026
Line 101:		Line 101:
	\|(12602123297335631, 80435758145817515, −80538738812075974)*		\|(12602123297335631, 80435758145817515, −80538738812075974)*
	\|https://oeis.org/A060467		\|https://oeis.org/A060467
			\|-
			\|352
			\|> 10<sup>80</sup>
			\|<math>\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} = 4</math>
			\|(<math> >4.373\times 10^{78}</math>, <math> >3.687\times 10^{79}</math>, <math> >1.544\times 10^{80}</math>)
			\|https://oeis.org/A369896
	\|}		\|}
	Note: For large champions (marked with * in Min Solution column) these are not known to be strictly the smallest magnitude solutions, instead they appear to be solutions with the minimal smallest value. If that is the case, then this still bounds BBdio since the score of these equations must be ≥ the minimal smallest variable value.		Note: For large champions (marked with * in Min Solution column) these are not known to be strictly the smallest magnitude solutions, instead they appear to be solutions with the minimal smallest value. If that is the case, then this still bounds BBdio since the score of these equations must be ≥ the minimal smallest variable value.

C7X at 02:21, 19 August 2026

2026-08-19T02:21:47Z

← Older revision		Revision as of 02:21, 19 August 2026
Line 1:		Line 1:
	A '''Diophantine equation''' is a polynomial equation with integer coefficients for which only integer solutions are sought. Diophantine equations are known to be Turing complete.		A '''Diophantine equation''' is a polynomial equation with integer coefficients for which only integer solutions are sought. Diophantine equations are known to be Turing complete via the [https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's_theorem MRDP theorem].

	== Definitions ==		== Definitions ==

Sligocki: /* Champions / Add note about exact minimal solution.

2026-08-14T16:38:37Z

Champions: Add * note about exact minimal solution.

← Older revision		Revision as of 16:38, 14 August 2026
Line 75:		Line 75:
	\|> 10<sup>6</sup>		\|> 10<sup>6</sup>
	\|<math>x^3 + x y^2 + x + z^3 - z + 1 = 0</math>		\|<math>x^3 + x y^2 + x + z^3 - z + 1 = 0</math>
	\|(-4280795, 4360815, 5427173)		\|(-4280795, 4360815, 5427173)*
	\|Andrew R. Booker via https://arxiv.org/abs/2108.08705 v3 page 23		\|Andrew R. Booker via https://arxiv.org/abs/2108.08705 v3 page 23
	\|-		\|-
Line 87:		Line 87:
	\|> 10<sup>10</sup>		\|> 10<sup>10</sup>
	\|<math>x^3 + y^3 + z^3 = 24</math>		\|<math>x^3 + y^3 + z^3 = 24</math>
	\|(-2901096694, -15550555555, 15584139827)		\|(-2901096694, -15550555555, 15584139827)*
	\|https://oeis.org/A060467		\|https://oeis.org/A060467
	\|-		\|-
Line 93:		Line 93:
	\|> 10<sup>15</sup>		\|> 10<sup>15</sup>
	\|<math>x^3 + y^3 + z^3 = 33</math>		\|<math>x^3 + y^3 + z^3 = 33</math>
	\|(-2736111468807040, -8778405442862239, 8866128975287528)		\|(-2736111468807040, -8778405442862239, 8866128975287528)*
	\|https://oeis.org/A060467		\|https://oeis.org/A060467
	\|-		\|-
Line 99:		Line 99:
	\|> 10<sup>16</sup>		\|> 10<sup>16</sup>
	\|<math>x^3 + y^3 + z^3 = 42</math>		\|<math>x^3 + y^3 + z^3 = 42</math>
	\|(12602123297335631, 80435758145817515, −80538738812075974)		\|(12602123297335631, 80435758145817515, −80538738812075974)*
	\|https://oeis.org/A060467		\|https://oeis.org/A060467
	\|}		\|}
			Note: For large champions (marked with * in Min Solution column) these are not known to be strictly the smallest magnitude solutions, instead they appear to be solutions with the minimal smallest value. If that is the case, then this still bounds BBdio since the score of these equations must be ≥ the minimal smallest variable value.

	== Cryptids ==		== Cryptids ==