Have the tides ever turned twice on any open problem?
$begingroup$
Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justification.
Some examples that come to mind:
Because many decision problems, such as graph non-isomorphism, have nice probabilistic protocols, i.e. they are in $mathsf{AM}$, but are not known to have certificates in $mathsf{NP}$, a reasonable conjecture was that $mathsf{NP}subsetmathsf{AM}$. However, based on the conjectured existence of strong-enough pseudorandom number generators, a reasonable statement nowadays is that $mathsf{NP}=mathsf{AM}$, etc.
I learned from Andrew Booker that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)in mathbb{Z}^3$ have varied, especially after some heuristics from Heath-Brown. It is reasonable to state that most $k$ have an infinite number of solutions.
Numerical evidence suggests that for all $x$, $y$, we have $pi(x+y)leq pi(x)+pi(y)$. This is commonly known as the "second Hardy-Littlewood Conjecture". See also this MSF question. However, a 1974 paper showed that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood.
- Number theory may also be littered with other such examples.
I'm interested if it has ever happened whether the process has ever repeated itself. That is:
Have there ever been situations wherein it is reasonable to suppose $P$, then, after some heuristic analysis, it is reasonable to supposed $neg P$, then, after further consideration, it is reasonable to suppose $P$?
I have read that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.
ho.history-overview
$endgroup$
|
show 5 more comments
$begingroup$
Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justification.
Some examples that come to mind:
Because many decision problems, such as graph non-isomorphism, have nice probabilistic protocols, i.e. they are in $mathsf{AM}$, but are not known to have certificates in $mathsf{NP}$, a reasonable conjecture was that $mathsf{NP}subsetmathsf{AM}$. However, based on the conjectured existence of strong-enough pseudorandom number generators, a reasonable statement nowadays is that $mathsf{NP}=mathsf{AM}$, etc.
I learned from Andrew Booker that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)in mathbb{Z}^3$ have varied, especially after some heuristics from Heath-Brown. It is reasonable to state that most $k$ have an infinite number of solutions.
Numerical evidence suggests that for all $x$, $y$, we have $pi(x+y)leq pi(x)+pi(y)$. This is commonly known as the "second Hardy-Littlewood Conjecture". See also this MSF question. However, a 1974 paper showed that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood.
- Number theory may also be littered with other such examples.
I'm interested if it has ever happened whether the process has ever repeated itself. That is:
Have there ever been situations wherein it is reasonable to suppose $P$, then, after some heuristic analysis, it is reasonable to supposed $neg P$, then, after further consideration, it is reasonable to suppose $P$?
I have read that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.
ho.history-overview
$endgroup$
1
$begingroup$
Fermat primes, boundedness of elliptic curve ranks
$endgroup$
– Wojowu
yesterday
$begingroup$
I have fixed some of what I think are typos, if I've got anything wrong feel free to edit back.
$endgroup$
– Wojowu
yesterday
6
$begingroup$
Not exactly an "open problem", but infinitesimals were fundamental in the initial development of calculus by Leibniz, then after Cauchy's rigorous development of analysis their use was considered non-rigorous, but then later in the 20th century a completely rigorous theory of infinitesimals was developed by Robinson.
$endgroup$
– Sam Hopkins
yesterday
$begingroup$
Regarding Booker's 1950s computing evidence: Wasn't it Swinnerton-Dyer who in the late 60s said something like "most number theory calculations to date do nothing more than pile up integers in the manner of a magpie"? (his work with Birch being an exception of course) $$ $$ For that matter, I don't see why the "prevailing opinion" should be for finiteness, indeed the 1955 Miller/Woollett paper (§14) perhaps suggests the opposite. OTOH, the later Gardiner/Lazarus Stein paper (1965) does opine that not all missing values would be represented, though they caution the reader about their evidence.
$endgroup$
– literature-searcher
yesterday
1
$begingroup$
@MarkS I think you could include examples where conjectures turned out to be wrong (once). In number theory, besides the Hardy-Littlewood example you mention there is the inequality $pi(x) < {rm Li}(x)$ (disproved by Littlewood with no example). For sums of 3 cubes, initially doubt was cast in writing on "$k notequiv 4, 5 bmod 9$" being sufficient for solvability (in the paper of Gardiner/Lazarus/Stein). So few thought about the problem in the 1950s that maybe those who did the numerical testing back then could be regarded as having the "prevailing opinion". :)
$endgroup$
– KConrad
5 hours ago
|
show 5 more comments
$begingroup$
Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justification.
Some examples that come to mind:
Because many decision problems, such as graph non-isomorphism, have nice probabilistic protocols, i.e. they are in $mathsf{AM}$, but are not known to have certificates in $mathsf{NP}$, a reasonable conjecture was that $mathsf{NP}subsetmathsf{AM}$. However, based on the conjectured existence of strong-enough pseudorandom number generators, a reasonable statement nowadays is that $mathsf{NP}=mathsf{AM}$, etc.
I learned from Andrew Booker that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)in mathbb{Z}^3$ have varied, especially after some heuristics from Heath-Brown. It is reasonable to state that most $k$ have an infinite number of solutions.
Numerical evidence suggests that for all $x$, $y$, we have $pi(x+y)leq pi(x)+pi(y)$. This is commonly known as the "second Hardy-Littlewood Conjecture". See also this MSF question. However, a 1974 paper showed that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood.
- Number theory may also be littered with other such examples.
I'm interested if it has ever happened whether the process has ever repeated itself. That is:
Have there ever been situations wherein it is reasonable to suppose $P$, then, after some heuristic analysis, it is reasonable to supposed $neg P$, then, after further consideration, it is reasonable to suppose $P$?
I have read that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.
ho.history-overview
$endgroup$
Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justification.
Some examples that come to mind:
Because many decision problems, such as graph non-isomorphism, have nice probabilistic protocols, i.e. they are in $mathsf{AM}$, but are not known to have certificates in $mathsf{NP}$, a reasonable conjecture was that $mathsf{NP}subsetmathsf{AM}$. However, based on the conjectured existence of strong-enough pseudorandom number generators, a reasonable statement nowadays is that $mathsf{NP}=mathsf{AM}$, etc.
I learned from Andrew Booker that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)in mathbb{Z}^3$ have varied, especially after some heuristics from Heath-Brown. It is reasonable to state that most $k$ have an infinite number of solutions.
Numerical evidence suggests that for all $x$, $y$, we have $pi(x+y)leq pi(x)+pi(y)$. This is commonly known as the "second Hardy-Littlewood Conjecture". See also this MSF question. However, a 1974 paper showed that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood.
- Number theory may also be littered with other such examples.
I'm interested if it has ever happened whether the process has ever repeated itself. That is:
Have there ever been situations wherein it is reasonable to suppose $P$, then, after some heuristic analysis, it is reasonable to supposed $neg P$, then, after further consideration, it is reasonable to suppose $P$?
I have read that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.
ho.history-overview
ho.history-overview
edited 5 hours ago
community wiki
6 revs, 2 users 96%
Mark S
1
$begingroup$
Fermat primes, boundedness of elliptic curve ranks
$endgroup$
– Wojowu
yesterday
$begingroup$
I have fixed some of what I think are typos, if I've got anything wrong feel free to edit back.
$endgroup$
– Wojowu
yesterday
6
$begingroup$
Not exactly an "open problem", but infinitesimals were fundamental in the initial development of calculus by Leibniz, then after Cauchy's rigorous development of analysis their use was considered non-rigorous, but then later in the 20th century a completely rigorous theory of infinitesimals was developed by Robinson.
$endgroup$
– Sam Hopkins
yesterday
$begingroup$
Regarding Booker's 1950s computing evidence: Wasn't it Swinnerton-Dyer who in the late 60s said something like "most number theory calculations to date do nothing more than pile up integers in the manner of a magpie"? (his work with Birch being an exception of course) $$ $$ For that matter, I don't see why the "prevailing opinion" should be for finiteness, indeed the 1955 Miller/Woollett paper (§14) perhaps suggests the opposite. OTOH, the later Gardiner/Lazarus Stein paper (1965) does opine that not all missing values would be represented, though they caution the reader about their evidence.
$endgroup$
– literature-searcher
yesterday
1
$begingroup$
@MarkS I think you could include examples where conjectures turned out to be wrong (once). In number theory, besides the Hardy-Littlewood example you mention there is the inequality $pi(x) < {rm Li}(x)$ (disproved by Littlewood with no example). For sums of 3 cubes, initially doubt was cast in writing on "$k notequiv 4, 5 bmod 9$" being sufficient for solvability (in the paper of Gardiner/Lazarus/Stein). So few thought about the problem in the 1950s that maybe those who did the numerical testing back then could be regarded as having the "prevailing opinion". :)
$endgroup$
– KConrad
5 hours ago
|
show 5 more comments
1
$begingroup$
Fermat primes, boundedness of elliptic curve ranks
$endgroup$
– Wojowu
yesterday
$begingroup$
I have fixed some of what I think are typos, if I've got anything wrong feel free to edit back.
$endgroup$
– Wojowu
yesterday
6
$begingroup$
Not exactly an "open problem", but infinitesimals were fundamental in the initial development of calculus by Leibniz, then after Cauchy's rigorous development of analysis their use was considered non-rigorous, but then later in the 20th century a completely rigorous theory of infinitesimals was developed by Robinson.
$endgroup$
– Sam Hopkins
yesterday
$begingroup$
Regarding Booker's 1950s computing evidence: Wasn't it Swinnerton-Dyer who in the late 60s said something like "most number theory calculations to date do nothing more than pile up integers in the manner of a magpie"? (his work with Birch being an exception of course) $$ $$ For that matter, I don't see why the "prevailing opinion" should be for finiteness, indeed the 1955 Miller/Woollett paper (§14) perhaps suggests the opposite. OTOH, the later Gardiner/Lazarus Stein paper (1965) does opine that not all missing values would be represented, though they caution the reader about their evidence.
$endgroup$
– literature-searcher
yesterday
1
$begingroup$
@MarkS I think you could include examples where conjectures turned out to be wrong (once). In number theory, besides the Hardy-Littlewood example you mention there is the inequality $pi(x) < {rm Li}(x)$ (disproved by Littlewood with no example). For sums of 3 cubes, initially doubt was cast in writing on "$k notequiv 4, 5 bmod 9$" being sufficient for solvability (in the paper of Gardiner/Lazarus/Stein). So few thought about the problem in the 1950s that maybe those who did the numerical testing back then could be regarded as having the "prevailing opinion". :)
$endgroup$
– KConrad
5 hours ago
1
1
$begingroup$
Fermat primes, boundedness of elliptic curve ranks
$endgroup$
– Wojowu
yesterday
$begingroup$
Fermat primes, boundedness of elliptic curve ranks
$endgroup$
– Wojowu
yesterday
$begingroup$
I have fixed some of what I think are typos, if I've got anything wrong feel free to edit back.
$endgroup$
– Wojowu
yesterday
$begingroup$
I have fixed some of what I think are typos, if I've got anything wrong feel free to edit back.
$endgroup$
– Wojowu
yesterday
6
6
$begingroup$
Not exactly an "open problem", but infinitesimals were fundamental in the initial development of calculus by Leibniz, then after Cauchy's rigorous development of analysis their use was considered non-rigorous, but then later in the 20th century a completely rigorous theory of infinitesimals was developed by Robinson.
$endgroup$
– Sam Hopkins
yesterday
$begingroup$
Not exactly an "open problem", but infinitesimals were fundamental in the initial development of calculus by Leibniz, then after Cauchy's rigorous development of analysis their use was considered non-rigorous, but then later in the 20th century a completely rigorous theory of infinitesimals was developed by Robinson.
$endgroup$
– Sam Hopkins
yesterday
$begingroup$
Regarding Booker's 1950s computing evidence: Wasn't it Swinnerton-Dyer who in the late 60s said something like "most number theory calculations to date do nothing more than pile up integers in the manner of a magpie"? (his work with Birch being an exception of course) $$ $$ For that matter, I don't see why the "prevailing opinion" should be for finiteness, indeed the 1955 Miller/Woollett paper (§14) perhaps suggests the opposite. OTOH, the later Gardiner/Lazarus Stein paper (1965) does opine that not all missing values would be represented, though they caution the reader about their evidence.
$endgroup$
– literature-searcher
yesterday
$begingroup$
Regarding Booker's 1950s computing evidence: Wasn't it Swinnerton-Dyer who in the late 60s said something like "most number theory calculations to date do nothing more than pile up integers in the manner of a magpie"? (his work with Birch being an exception of course) $$ $$ For that matter, I don't see why the "prevailing opinion" should be for finiteness, indeed the 1955 Miller/Woollett paper (§14) perhaps suggests the opposite. OTOH, the later Gardiner/Lazarus Stein paper (1965) does opine that not all missing values would be represented, though they caution the reader about their evidence.
$endgroup$
– literature-searcher
yesterday
1
1
$begingroup$
@MarkS I think you could include examples where conjectures turned out to be wrong (once). In number theory, besides the Hardy-Littlewood example you mention there is the inequality $pi(x) < {rm Li}(x)$ (disproved by Littlewood with no example). For sums of 3 cubes, initially doubt was cast in writing on "$k notequiv 4, 5 bmod 9$" being sufficient for solvability (in the paper of Gardiner/Lazarus/Stein). So few thought about the problem in the 1950s that maybe those who did the numerical testing back then could be regarded as having the "prevailing opinion". :)
$endgroup$
– KConrad
5 hours ago
$begingroup$
@MarkS I think you could include examples where conjectures turned out to be wrong (once). In number theory, besides the Hardy-Littlewood example you mention there is the inequality $pi(x) < {rm Li}(x)$ (disproved by Littlewood with no example). For sums of 3 cubes, initially doubt was cast in writing on "$k notequiv 4, 5 bmod 9$" being sufficient for solvability (in the paper of Gardiner/Lazarus/Stein). So few thought about the problem in the 1950s that maybe those who did the numerical testing back then could be regarded as having the "prevailing opinion". :)
$endgroup$
– KConrad
5 hours ago
|
show 5 more comments
3 Answers
3
active
oldest
votes
$begingroup$
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/mathbb{Q}$, both the size of the torsion subgroup of $E(mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $Bbb Q$?), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: Conjecture There are only finitely many $E/mathbb{Q}$ for which the rank of $E(mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)
$endgroup$
1
$begingroup$
As is also the case for some other examples discussed, this instance could eventually become an example of a tide that turned (at least) thrice :)
$endgroup$
– Nell
yesterday
$begingroup$
The last time I checked the status on Elkie's curve is still that the rank is at least $28$, but not known to be exact. Has that changed?
$endgroup$
– Stanley Yao Xiao
yesterday
1
$begingroup$
@StanleyYaoXiao You're correct, I'm just being my usual imprecise self. At present, there seems to be no hope of doing any sort of descent to show that the rank is exactly 28.
$endgroup$
– Joe Silverman
yesterday
5
$begingroup$
@StanleyYaoXiao The rank is 28 under GRH arxiv.org/abs/1606.07178
$endgroup$
– Aurel
23 hours ago
$begingroup$
I hadn't heard about an initial belief in boundedness of ranks---are there some remarks about this in older literature?
$endgroup$
– Kimball
7 hours ago
|
show 1 more comment
$begingroup$
$P=,!$ Calabi’s conjecture.
Specifically, the link says “By the late 1960s, many were doubtful of the Calabi conjecture”, then Yau did “produce a "counterexample" to the conjecture. The "counterexample" was soon discovered to be flawed”, finally in 1976 Yau proved the conjecture. More details: Yau’s autobiographical account.
$endgroup$
2
$begingroup$
(Francois Ziegler is the OP. I merely incorporated a comment he made into the answer.)
$endgroup$
– Todd Trimble♦
12 hours ago
add a comment |
$begingroup$
I think the Busemann-Petty problem is an example like what you're asking, although changes in opinion would be due to progress (positive and negative) rather than any heuristic analysis.
A great description of the problem and what happened over time is the top answer at Widely accepted mathematical results that were later shown to be wrong?. See also https://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem.
It's a geometric conjecture (well, a "question" but I think it's easier to follow if I call it a conjecture) that is true in 2 dimensions and everyone expected it to hold in all higher dimensions, but things turned out differently with a surprise ending. Over time counterexamples were found to the conjecture in every dimension above 5, then it was proved to be true in dimension 3, then a counterexample was found in dimension 4, then the counterexample in dimension 4 was shown to be an example rather than a counterexample, and then it was proved to be true in dimension 4 by the same person who earlier gave the "counterexample" in dimension 4, with both of his papers appearing in the Annals.
$endgroup$
add a comment |
protected by Carlo Beenakker 10 hours ago
Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead?
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/mathbb{Q}$, both the size of the torsion subgroup of $E(mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $Bbb Q$?), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: Conjecture There are only finitely many $E/mathbb{Q}$ for which the rank of $E(mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)
$endgroup$
1
$begingroup$
As is also the case for some other examples discussed, this instance could eventually become an example of a tide that turned (at least) thrice :)
$endgroup$
– Nell
yesterday
$begingroup$
The last time I checked the status on Elkie's curve is still that the rank is at least $28$, but not known to be exact. Has that changed?
$endgroup$
– Stanley Yao Xiao
yesterday
1
$begingroup$
@StanleyYaoXiao You're correct, I'm just being my usual imprecise self. At present, there seems to be no hope of doing any sort of descent to show that the rank is exactly 28.
$endgroup$
– Joe Silverman
yesterday
5
$begingroup$
@StanleyYaoXiao The rank is 28 under GRH arxiv.org/abs/1606.07178
$endgroup$
– Aurel
23 hours ago
$begingroup$
I hadn't heard about an initial belief in boundedness of ranks---are there some remarks about this in older literature?
$endgroup$
– Kimball
7 hours ago
|
show 1 more comment
$begingroup$
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/mathbb{Q}$, both the size of the torsion subgroup of $E(mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $Bbb Q$?), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: Conjecture There are only finitely many $E/mathbb{Q}$ for which the rank of $E(mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)
$endgroup$
1
$begingroup$
As is also the case for some other examples discussed, this instance could eventually become an example of a tide that turned (at least) thrice :)
$endgroup$
– Nell
yesterday
$begingroup$
The last time I checked the status on Elkie's curve is still that the rank is at least $28$, but not known to be exact. Has that changed?
$endgroup$
– Stanley Yao Xiao
yesterday
1
$begingroup$
@StanleyYaoXiao You're correct, I'm just being my usual imprecise self. At present, there seems to be no hope of doing any sort of descent to show that the rank is exactly 28.
$endgroup$
– Joe Silverman
yesterday
5
$begingroup$
@StanleyYaoXiao The rank is 28 under GRH arxiv.org/abs/1606.07178
$endgroup$
– Aurel
23 hours ago
$begingroup$
I hadn't heard about an initial belief in boundedness of ranks---are there some remarks about this in older literature?
$endgroup$
– Kimball
7 hours ago
|
show 1 more comment
$begingroup$
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/mathbb{Q}$, both the size of the torsion subgroup of $E(mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $Bbb Q$?), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: Conjecture There are only finitely many $E/mathbb{Q}$ for which the rank of $E(mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)
$endgroup$
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/mathbb{Q}$, both the size of the torsion subgroup of $E(mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $Bbb Q$?), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: Conjecture There are only finitely many $E/mathbb{Q}$ for which the rank of $E(mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)
edited 7 hours ago
community wiki
2 revs, 2 users 67%
Joe Silverman
1
$begingroup$
As is also the case for some other examples discussed, this instance could eventually become an example of a tide that turned (at least) thrice :)
$endgroup$
– Nell
yesterday
$begingroup$
The last time I checked the status on Elkie's curve is still that the rank is at least $28$, but not known to be exact. Has that changed?
$endgroup$
– Stanley Yao Xiao
yesterday
1
$begingroup$
@StanleyYaoXiao You're correct, I'm just being my usual imprecise self. At present, there seems to be no hope of doing any sort of descent to show that the rank is exactly 28.
$endgroup$
– Joe Silverman
yesterday
5
$begingroup$
@StanleyYaoXiao The rank is 28 under GRH arxiv.org/abs/1606.07178
$endgroup$
– Aurel
23 hours ago
$begingroup$
I hadn't heard about an initial belief in boundedness of ranks---are there some remarks about this in older literature?
$endgroup$
– Kimball
7 hours ago
|
show 1 more comment
1
$begingroup$
As is also the case for some other examples discussed, this instance could eventually become an example of a tide that turned (at least) thrice :)
$endgroup$
– Nell
yesterday
$begingroup$
The last time I checked the status on Elkie's curve is still that the rank is at least $28$, but not known to be exact. Has that changed?
$endgroup$
– Stanley Yao Xiao
yesterday
1
$begingroup$
@StanleyYaoXiao You're correct, I'm just being my usual imprecise self. At present, there seems to be no hope of doing any sort of descent to show that the rank is exactly 28.
$endgroup$
– Joe Silverman
yesterday
5
$begingroup$
@StanleyYaoXiao The rank is 28 under GRH arxiv.org/abs/1606.07178
$endgroup$
– Aurel
23 hours ago
$begingroup$
I hadn't heard about an initial belief in boundedness of ranks---are there some remarks about this in older literature?
$endgroup$
– Kimball
7 hours ago
1
1
$begingroup$
As is also the case for some other examples discussed, this instance could eventually become an example of a tide that turned (at least) thrice :)
$endgroup$
– Nell
yesterday
$begingroup$
As is also the case for some other examples discussed, this instance could eventually become an example of a tide that turned (at least) thrice :)
$endgroup$
– Nell
yesterday
$begingroup$
The last time I checked the status on Elkie's curve is still that the rank is at least $28$, but not known to be exact. Has that changed?
$endgroup$
– Stanley Yao Xiao
yesterday
$begingroup$
The last time I checked the status on Elkie's curve is still that the rank is at least $28$, but not known to be exact. Has that changed?
$endgroup$
– Stanley Yao Xiao
yesterday
1
1
$begingroup$
@StanleyYaoXiao You're correct, I'm just being my usual imprecise self. At present, there seems to be no hope of doing any sort of descent to show that the rank is exactly 28.
$endgroup$
– Joe Silverman
yesterday
$begingroup$
@StanleyYaoXiao You're correct, I'm just being my usual imprecise self. At present, there seems to be no hope of doing any sort of descent to show that the rank is exactly 28.
$endgroup$
– Joe Silverman
yesterday
5
5
$begingroup$
@StanleyYaoXiao The rank is 28 under GRH arxiv.org/abs/1606.07178
$endgroup$
– Aurel
23 hours ago
$begingroup$
@StanleyYaoXiao The rank is 28 under GRH arxiv.org/abs/1606.07178
$endgroup$
– Aurel
23 hours ago
$begingroup$
I hadn't heard about an initial belief in boundedness of ranks---are there some remarks about this in older literature?
$endgroup$
– Kimball
7 hours ago
$begingroup$
I hadn't heard about an initial belief in boundedness of ranks---are there some remarks about this in older literature?
$endgroup$
– Kimball
7 hours ago
|
show 1 more comment
$begingroup$
$P=,!$ Calabi’s conjecture.
Specifically, the link says “By the late 1960s, many were doubtful of the Calabi conjecture”, then Yau did “produce a "counterexample" to the conjecture. The "counterexample" was soon discovered to be flawed”, finally in 1976 Yau proved the conjecture. More details: Yau’s autobiographical account.
$endgroup$
2
$begingroup$
(Francois Ziegler is the OP. I merely incorporated a comment he made into the answer.)
$endgroup$
– Todd Trimble♦
12 hours ago
add a comment |
$begingroup$
$P=,!$ Calabi’s conjecture.
Specifically, the link says “By the late 1960s, many were doubtful of the Calabi conjecture”, then Yau did “produce a "counterexample" to the conjecture. The "counterexample" was soon discovered to be flawed”, finally in 1976 Yau proved the conjecture. More details: Yau’s autobiographical account.
$endgroup$
2
$begingroup$
(Francois Ziegler is the OP. I merely incorporated a comment he made into the answer.)
$endgroup$
– Todd Trimble♦
12 hours ago
add a comment |
$begingroup$
$P=,!$ Calabi’s conjecture.
Specifically, the link says “By the late 1960s, many were doubtful of the Calabi conjecture”, then Yau did “produce a "counterexample" to the conjecture. The "counterexample" was soon discovered to be flawed”, finally in 1976 Yau proved the conjecture. More details: Yau’s autobiographical account.
$endgroup$
$P=,!$ Calabi’s conjecture.
Specifically, the link says “By the late 1960s, many were doubtful of the Calabi conjecture”, then Yau did “produce a "counterexample" to the conjecture. The "counterexample" was soon discovered to be flawed”, finally in 1976 Yau proved the conjecture. More details: Yau’s autobiographical account.
edited yesterday
community wiki
2 revs, 2 users 60%
Todd Trimble
2
$begingroup$
(Francois Ziegler is the OP. I merely incorporated a comment he made into the answer.)
$endgroup$
– Todd Trimble♦
12 hours ago
add a comment |
2
$begingroup$
(Francois Ziegler is the OP. I merely incorporated a comment he made into the answer.)
$endgroup$
– Todd Trimble♦
12 hours ago
2
2
$begingroup$
(Francois Ziegler is the OP. I merely incorporated a comment he made into the answer.)
$endgroup$
– Todd Trimble♦
12 hours ago
$begingroup$
(Francois Ziegler is the OP. I merely incorporated a comment he made into the answer.)
$endgroup$
– Todd Trimble♦
12 hours ago
add a comment |
$begingroup$
I think the Busemann-Petty problem is an example like what you're asking, although changes in opinion would be due to progress (positive and negative) rather than any heuristic analysis.
A great description of the problem and what happened over time is the top answer at Widely accepted mathematical results that were later shown to be wrong?. See also https://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem.
It's a geometric conjecture (well, a "question" but I think it's easier to follow if I call it a conjecture) that is true in 2 dimensions and everyone expected it to hold in all higher dimensions, but things turned out differently with a surprise ending. Over time counterexamples were found to the conjecture in every dimension above 5, then it was proved to be true in dimension 3, then a counterexample was found in dimension 4, then the counterexample in dimension 4 was shown to be an example rather than a counterexample, and then it was proved to be true in dimension 4 by the same person who earlier gave the "counterexample" in dimension 4, with both of his papers appearing in the Annals.
$endgroup$
add a comment |
$begingroup$
I think the Busemann-Petty problem is an example like what you're asking, although changes in opinion would be due to progress (positive and negative) rather than any heuristic analysis.
A great description of the problem and what happened over time is the top answer at Widely accepted mathematical results that were later shown to be wrong?. See also https://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem.
It's a geometric conjecture (well, a "question" but I think it's easier to follow if I call it a conjecture) that is true in 2 dimensions and everyone expected it to hold in all higher dimensions, but things turned out differently with a surprise ending. Over time counterexamples were found to the conjecture in every dimension above 5, then it was proved to be true in dimension 3, then a counterexample was found in dimension 4, then the counterexample in dimension 4 was shown to be an example rather than a counterexample, and then it was proved to be true in dimension 4 by the same person who earlier gave the "counterexample" in dimension 4, with both of his papers appearing in the Annals.
$endgroup$
add a comment |
$begingroup$
I think the Busemann-Petty problem is an example like what you're asking, although changes in opinion would be due to progress (positive and negative) rather than any heuristic analysis.
A great description of the problem and what happened over time is the top answer at Widely accepted mathematical results that were later shown to be wrong?. See also https://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem.
It's a geometric conjecture (well, a "question" but I think it's easier to follow if I call it a conjecture) that is true in 2 dimensions and everyone expected it to hold in all higher dimensions, but things turned out differently with a surprise ending. Over time counterexamples were found to the conjecture in every dimension above 5, then it was proved to be true in dimension 3, then a counterexample was found in dimension 4, then the counterexample in dimension 4 was shown to be an example rather than a counterexample, and then it was proved to be true in dimension 4 by the same person who earlier gave the "counterexample" in dimension 4, with both of his papers appearing in the Annals.
$endgroup$
I think the Busemann-Petty problem is an example like what you're asking, although changes in opinion would be due to progress (positive and negative) rather than any heuristic analysis.
A great description of the problem and what happened over time is the top answer at Widely accepted mathematical results that were later shown to be wrong?. See also https://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem.
It's a geometric conjecture (well, a "question" but I think it's easier to follow if I call it a conjecture) that is true in 2 dimensions and everyone expected it to hold in all higher dimensions, but things turned out differently with a surprise ending. Over time counterexamples were found to the conjecture in every dimension above 5, then it was proved to be true in dimension 3, then a counterexample was found in dimension 4, then the counterexample in dimension 4 was shown to be an example rather than a counterexample, and then it was proved to be true in dimension 4 by the same person who earlier gave the "counterexample" in dimension 4, with both of his papers appearing in the Annals.
edited 6 hours ago
community wiki
2 revs
KConrad
add a comment |
add a comment |
protected by Carlo Beenakker 10 hours ago
Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead?
1
$begingroup$
Fermat primes, boundedness of elliptic curve ranks
$endgroup$
– Wojowu
yesterday
$begingroup$
I have fixed some of what I think are typos, if I've got anything wrong feel free to edit back.
$endgroup$
– Wojowu
yesterday
6
$begingroup$
Not exactly an "open problem", but infinitesimals were fundamental in the initial development of calculus by Leibniz, then after Cauchy's rigorous development of analysis their use was considered non-rigorous, but then later in the 20th century a completely rigorous theory of infinitesimals was developed by Robinson.
$endgroup$
– Sam Hopkins
yesterday
$begingroup$
Regarding Booker's 1950s computing evidence: Wasn't it Swinnerton-Dyer who in the late 60s said something like "most number theory calculations to date do nothing more than pile up integers in the manner of a magpie"? (his work with Birch being an exception of course) $$ $$ For that matter, I don't see why the "prevailing opinion" should be for finiteness, indeed the 1955 Miller/Woollett paper (§14) perhaps suggests the opposite. OTOH, the later Gardiner/Lazarus Stein paper (1965) does opine that not all missing values would be represented, though they caution the reader about their evidence.
$endgroup$
– literature-searcher
yesterday
1
$begingroup$
@MarkS I think you could include examples where conjectures turned out to be wrong (once). In number theory, besides the Hardy-Littlewood example you mention there is the inequality $pi(x) < {rm Li}(x)$ (disproved by Littlewood with no example). For sums of 3 cubes, initially doubt was cast in writing on "$k notequiv 4, 5 bmod 9$" being sufficient for solvability (in the paper of Gardiner/Lazarus/Stein). So few thought about the problem in the 1950s that maybe those who did the numerical testing back then could be regarded as having the "prevailing opinion". :)
$endgroup$
– KConrad
5 hours ago