Revisions to Which integers can be expressed as a sum of three cubes in infinitely many ways?

165 = (-385495523231271884)^3 + 383344975542639445^3 + 98422560467622814^3

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edited Oct 3, 2019 at 13:20

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As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = -15250^3 + 13139^3 + 10853^3$$96 = 13139^3 -15250^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = -112969^³ + 103532^³ + 69241^³$$80 = 103532^3 -112969^3 + 69241^3$

(1990s)

$39 = -159380^³ + 134476^³ + 117367^³$$39 = 134476^3 - 159380^3 + 117367^3$
$75 = – 435203231^³ + 435203083^³ + 4381159^³$$75 = 435203083^3 – 435203231^3 + 4381159^3$
$84 = 41639611^³ – 41531726^³ – 8241191^³$$84 = 41639611^3 – 41531726^3 – 8241191^3$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = -61922712865^3 + 23961292454^³ + 60702901317^3$$52 = 23961292454^3 - 61922712865^3 + 60702901317^3$
$74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$$74 = 66229832190556^3 − 284650292555885^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3$$906 = 72054089679353378^3 -74924259395610397^3 + 35961979615356503^3$

$165 = 383344975542639445^3 -385495523231271884^3 + 98422560467622814^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $390$, $579$, $627$, $633$, $732$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = -15250^3 + 13139^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = -112969^³ + 103532^³ + 69241^³$

(1990s)

$39 = -159380^³ + 134476^³ + 117367^³$
$75 = – 435203231^³ + 435203083^³ + 4381159^³$
$84 = 41639611^³ – 41531726^³ – 8241191^³$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = -61922712865^3 + 23961292454^³ + 60702901317^3$
$74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $390$, $579$, $627$, $633$, $732$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = 13139^3 -15250^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = 103532^3 -112969^3 + 69241^3$

(1990s)

$39 = 134476^3 - 159380^3 + 117367^3$
$75 = 435203083^3 – 435203231^3 + 4381159^3$
$84 = 41639611^3 – 41531726^3 – 8241191^3$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = 23961292454^3 - 61922712865^3 + 60702901317^3$
$74 = 66229832190556^3 − 284650292555885^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = 72054089679353378^3 -74924259395610397^3 + 35961979615356503^3$

$165 = 383344975542639445^3 -385495523231271884^3 + 98422560467622814^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $390$, $579$, $627$, $633$, $732$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

165 = (-385495523231271884)^3 + 383344975542639445^3 + 98422560467622814^3

Source Link

edit approved Oct 3, 2019 at 13:20

Robin Houston

2.9k
1
31
54

As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = -15250^3 + 13139^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = -112969^³ + 103532^³ + 69241^³$

(1990s)

$39 = -159380^³ + 134476^³ + 117367^³$
$75 = – 435203231^³ + 435203083^³ + 4381159^³$
$84 = 41639611^³ – 41531726^³ – 8241191^³$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = -61922712865^3 + 23961292454^³ + 60702901317^3$
$74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $165$, $390$, $579$, $627$, $633$, $732$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = -15250^3 + 13139^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = -112969^³ + 103532^³ + 69241^³$

(1990s)

$39 = -159380^³ + 134476^³ + 117367^³$
$75 = – 435203231^³ + 435203083^³ + 4381159^³$
$84 = 41639611^³ – 41531726^³ – 8241191^³$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = -61922712865^3 + 23961292454^³ + 60702901317^3$
$74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $165$, $390$, $579$, $627$, $633$, $732$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = -15250^3 + 13139^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = -112969^³ + 103532^³ + 69241^³$

(1990s)

$39 = -159380^³ + 134476^³ + 117367^³$
$75 = – 435203231^³ + 435203083^³ + 4381159^³$
$84 = 41639611^³ – 41531726^³ – 8241191^³$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = -61922712865^3 + 23961292454^³ + 60702901317^3$
$74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $390$, $579$, $627$, $633$, $732$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

The result for 906 is documented on https://math.mit.edu/~drew/, credited as joint work with Andrew Booker

Source Link

edit approved Sep 23, 2019 at 10:57

Robin Houston

2.9k
1
31
54

As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = -15250^3 + 13139^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = -112969^³ + 103532^³ + 69241^³$

(1990s)

$39 = -159380^³ + 134476^³ + 117367^³$
$75 = – 435203231^³ + 435203083^³ + 4381159^³$
$84 = 41639611^³ – 41531726^³ – 8241191^³$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = -61922712865^3 + 23961292454^³ + 60702901317^3$
$74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $165$, $390$, $579$, $627$, $633$, $732$, $906$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = -15250^3 + 13139^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = -112969^³ + 103532^³ + 69241^³$

(1990s)

$39 = -159380^³ + 134476^³ + 117367^³$
$75 = – 435203231^³ + 435203083^³ + 4381159^³$
$84 = 41639611^³ – 41531726^³ – 8241191^³$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = -61922712865^3 + 23961292454^³ + 60702901317^3$
$74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $165$, $390$, $579$, $627$, $633$, $732$, $906$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.
Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

$87 = 4271^3 – 4126^3 – 1972^3$
$96 = -15250^3 + 13139^3 + 10853^3$
$91 = 83538^3 – 67134^3 – 65453^3$
$80 = -112969^³ + 103532^³ + 69241^³$

(1990s)

$39 = -159380^³ + 134476^³ + 117367^³$
$75 = – 435203231^³ + 435203083^³ + 4381159^³$
$84 = 41639611^³ – 41531726^³ – 8241191^³$

(2000s)

$30 = 2220422932^3 – 2218888517^3 – 283059965^3$
$52 = -61922712865^3 + 23961292454^³ + 60702901317^3$
$74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$

(2019)

$33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
$42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
$3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$
$906 = (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3$

Remark: for $n \le 1000$, the problem is still open only for $114$, $165$, $390$, $579$, $627$, $633$, $732$, $921$, and $975$ (see this paper and this paper, and also this).