Combinatorics problems that can be solved via infinite descent The 2019 Stack Overflow Developer Survey Results Are InWhat is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $fracm+1n+fracn+1m$ is a positive integerProof of the infinite descent principleBooks for high school students starting on college mathPuzzles or short exercises illustrating mathematical problem solving to freshman studentsFermat solved $x^2+2=y^3$ by infinite descent?Outline for high school combinatorics class?Proofs by infinite descent on the number of prime factors of an integerCombinatorics and Discrete Mathematics: problems with solutions(set theory, congruences,Recurrence relation…)Combinatorial Species, significance and problems can be solved with it.reference-Request-IMOCombinatorial problems that were solved using the representation theory of finite groups?Good books to learn olympiad geometry,number theory, combinatorics and more
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Combinatorics problems that can be solved via infinite descent
The 2019 Stack Overflow Developer Survey Results Are InWhat is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $fracm+1n+fracn+1m$ is a positive integerProof of the infinite descent principleBooks for high school students starting on college mathPuzzles or short exercises illustrating mathematical problem solving to freshman studentsFermat solved $x^2+2=y^3$ by infinite descent?Outline for high school combinatorics class?Proofs by infinite descent on the number of prime factors of an integerCombinatorics and Discrete Mathematics: problems with solutions(set theory, congruences,Recurrence relation…)Combinatorial Species, significance and problems can be solved with it.reference-Request-IMOCombinatorial problems that were solved using the representation theory of finite groups?Good books to learn olympiad geometry,number theory, combinatorics and more
$begingroup$
I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a problem(s) from combinatorics and/or any other field of mathematics. Here are some problems from number theory:
Prove that a following equations have no nontrivial solutions in $mathbbZ$:
$a^3+2b^3 = 4c^3$
$2a^2+3b^2 = c^2+6d^2$
$x^2 + y^2 + z^2 = 2xyz$
$x^4+y^4 = z^2$
combinatorics number-theory big-list infinite-descent
asked Apr 6 at 18:41
Maria MazurMaria Mazur
49.9k1361125
$endgroup$
add a comment |
$begingroup$
I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a problem(s) from combinatorics and/or any other field of mathematics. Here are some problems from number theory:
Prove that a following equations have no nontrivial solutions in $mathbbZ$:
$a^3+2b^3 = 4c^3$
$2a^2+3b^2 = c^2+6d^2$
$x^2 + y^2 + z^2 = 2xyz$
$x^4+y^4 = z^2$
combinatorics number-theory big-list infinite-descent
asked Apr 6 at 18:41
Maria MazurMaria Mazur
49.9k1361125
$endgroup$
1
$begingroup$
I wonder why should this be closed?
$endgroup$
– Maria Mazur
Apr 6 at 20:30
2
$begingroup$
FWIW I voted to leave this open. However, I do have the misgiving that this type of a question does not have a "correct" answer. I guess opinions differ whether that makes a question unsuitable. It sounds like you welcome many answers, so may be using the "meta"tag big-list would be a good idea?
$endgroup$
– Jyrki Lahtonen
Apr 6 at 21:03
add a comment |
$begingroup$
I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a problem(s) from combinatorics and/or any other field of mathematics. Here are some problems from number theory:
Prove that a following equations have no nontrivial solutions in $mathbbZ$:
$a^3+2b^3 = 4c^3$
$2a^2+3b^2 = c^2+6d^2$
$x^2 + y^2 + z^2 = 2xyz$
$x^4+y^4 = z^2$
combinatorics number-theory big-list infinite-descent
asked Apr 6 at 18:41
Maria MazurMaria Mazur
49.9k1361125
$endgroup$
I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a problem(s) from combinatorics and/or any other field of mathematics. Here are some problems from number theory:
Prove that a following equations have no nontrivial solutions in $mathbbZ$:
$a^3+2b^3 = 4c^3$
$2a^2+3b^2 = c^2+6d^2$
$x^2 + y^2 + z^2 = 2xyz$
$x^4+y^4 = z^2$
combinatorics number-theory big-list infinite-descent
combinatorics number-theory big-list infinite-descent
asked Apr 6 at 18:41
Maria MazurMaria Mazur
49.9k1361125
asked Apr 6 at 18:41
Maria MazurMaria Mazur
49.9k1361125
edited Apr 6 at 21:23
Maria Mazur
asked Apr 6 at 18:41
Maria MazurMaria Mazur
49.9k1361125
asked Apr 6 at 18:41
Maria MazurMaria Mazur
49.9k1361125
asked Apr 6 at 18:41
Maria MazurMaria Mazur
49.9k1361125
49.9k1361125
1
$begingroup$
I wonder why should this be closed?
$endgroup$
– Maria Mazur
Apr 6 at 20:30
2
$begingroup$
FWIW I voted to leave this open. However, I do have the misgiving that this type of a question does not have a "correct" answer. I guess opinions differ whether that makes a question unsuitable. It sounds like you welcome many answers, so may be using the "meta"tag big-list would be a good idea?
$endgroup$
– Jyrki Lahtonen
Apr 6 at 21:03
add a comment |
1
$begingroup$
I wonder why should this be closed?
$endgroup$
– Maria Mazur
Apr 6 at 20:30
2
$begingroup$
FWIW I voted to leave this open. However, I do have the misgiving that this type of a question does not have a "correct" answer. I guess opinions differ whether that makes a question unsuitable. It sounds like you welcome many answers, so may be using the "meta"tag big-list would be a good idea?
$endgroup$
– Jyrki Lahtonen
Apr 6 at 21:03
1
1
$begingroup$
I wonder why should this be closed?
$endgroup$
– Maria Mazur
Apr 6 at 20:30
$begingroup$
I wonder why should this be closed?
$endgroup$
– Maria Mazur
Apr 6 at 20:30
2
2
$begingroup$
FWIW I voted to leave this open. However, I do have the misgiving that this type of a question does not have a "correct" answer. I guess opinions differ whether that makes a question unsuitable. It sounds like you welcome many answers, so may be using the "meta"tag big-list would be a good idea?
$endgroup$
– Jyrki Lahtonen
Apr 6 at 21:03
$begingroup$
FWIW I voted to leave this open. However, I do have the misgiving that this type of a question does not have a "correct" answer. I guess opinions differ whether that makes a question unsuitable. It sounds like you welcome many answers, so may be using the "meta"tag big-list would be a good idea?
$endgroup$
– Jyrki Lahtonen
Apr 6 at 21:03
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
What about this one :
Let $(a,b,c)$ in $mathbbN$ such that $(a^2+b^2)/(1+ab) =c $
Prove that $c = p^2$ with $p in mathbbN$
I don't have a proof though...
answered Apr 6 at 18:48
FlorianFlorian
22614
$endgroup$
$begingroup$
This one can be done by ID?
$endgroup$
– Maria Mazur
Apr 6 at 18:50
1
$begingroup$
They say so here les-mathematiques.net/phorum/read.php?5,339463,339884 (in French)
$endgroup$
– Florian
Apr 6 at 18:51
1
$begingroup$
@DonThousand see zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf
$endgroup$
– Will Jagy
Apr 6 at 19:29
2
$begingroup$
@DonThousand No, syymetries governing descent in the group of integer points on concis is much older - see the links I have here.
$endgroup$
– Bill Dubuque
Apr 6 at 20:02
1
$begingroup$
@MariaMazur For a much deeper understanding of so-called "Vieta jumping" see the literature I cite in my prior comment.
$endgroup$
– Bill Dubuque
Apr 6 at 20:04
|
show 5 more comments
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
What about this one :
Let $(a,b,c)$ in $mathbbN$ such that $(a^2+b^2)/(1+ab) =c $
Prove that $c = p^2$ with $p in mathbbN$
I don't have a proof though...
answered Apr 6 at 18:48
FlorianFlorian
22614
$endgroup$
$begingroup$
This one can be done by ID?
$endgroup$
– Maria Mazur
Apr 6 at 18:50
1
$begingroup$
They say so here les-mathematiques.net/phorum/read.php?5,339463,339884 (in French)
$endgroup$
– Florian
Apr 6 at 18:51
1
$begingroup$
@DonThousand see zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf
$endgroup$
– Will Jagy
Apr 6 at 19:29
2
$begingroup$
@DonThousand No, syymetries governing descent in the group of integer points on concis is much older - see the links I have here.
$endgroup$
– Bill Dubuque
Apr 6 at 20:02
1
$begingroup$
@MariaMazur For a much deeper understanding of so-called "Vieta jumping" see the literature I cite in my prior comment.
$endgroup$
– Bill Dubuque
Apr 6 at 20:04
|
show 5 more comments
$begingroup$
What about this one :
Let $(a,b,c)$ in $mathbbN$ such that $(a^2+b^2)/(1+ab) =c $
Prove that $c = p^2$ with $p in mathbbN$
I don't have a proof though...
answered Apr 6 at 18:48
FlorianFlorian
22614
$endgroup$
$begingroup$
This one can be done by ID?
$endgroup$
– Maria Mazur
Apr 6 at 18:50
1
$begingroup$
They say so here les-mathematiques.net/phorum/read.php?5,339463,339884 (in French)
$endgroup$
– Florian
Apr 6 at 18:51
1
$begingroup$
@DonThousand see zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf
$endgroup$
– Will Jagy
Apr 6 at 19:29
2
$begingroup$
@DonThousand No, syymetries governing descent in the group of integer points on concis is much older - see the links I have here.
$endgroup$
– Bill Dubuque
Apr 6 at 20:02
1
$begingroup$
@MariaMazur For a much deeper understanding of so-called "Vieta jumping" see the literature I cite in my prior comment.
$endgroup$
– Bill Dubuque
Apr 6 at 20:04
|
show 5 more comments
$begingroup$
What about this one :
Let $(a,b,c)$ in $mathbbN$ such that $(a^2+b^2)/(1+ab) =c $
Prove that $c = p^2$ with $p in mathbbN$
I don't have a proof though...
answered Apr 6 at 18:48
FlorianFlorian
22614
$endgroup$
What about this one :
Let $(a,b,c)$ in $mathbbN$ such that $(a^2+b^2)/(1+ab) =c $
Prove that $c = p^2$ with $p in mathbbN$
I don't have a proof though...
answered Apr 6 at 18:48
FlorianFlorian
22614
answered Apr 6 at 18:48
FlorianFlorian
22614
answered Apr 6 at 18:48
FlorianFlorian
22614
answered Apr 6 at 18:48
FlorianFlorian
22614
22614
$begingroup$
This one can be done by ID?
$endgroup$
– Maria Mazur
Apr 6 at 18:50
1
$begingroup$
They say so here les-mathematiques.net/phorum/read.php?5,339463,339884 (in French)
$endgroup$
– Florian
Apr 6 at 18:51
1
$begingroup$
@DonThousand see zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf
$endgroup$
– Will Jagy
Apr 6 at 19:29
2
$begingroup$
@DonThousand No, syymetries governing descent in the group of integer points on concis is much older - see the links I have here.
$endgroup$
– Bill Dubuque
Apr 6 at 20:02
1
$begingroup$
@MariaMazur For a much deeper understanding of so-called "Vieta jumping" see the literature I cite in my prior comment.
$endgroup$
– Bill Dubuque
Apr 6 at 20:04
|
show 5 more comments
$begingroup$
This one can be done by ID?
$endgroup$
– Maria Mazur
Apr 6 at 18:50
1
$begingroup$
They say so here les-mathematiques.net/phorum/read.php?5,339463,339884 (in French)
$endgroup$
– Florian
Apr 6 at 18:51
1
$begingroup$
@DonThousand see zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf
$endgroup$
– Will Jagy
Apr 6 at 19:29
2
$begingroup$
@DonThousand No, syymetries governing descent in the group of integer points on concis is much older - see the links I have here.
$endgroup$
– Bill Dubuque
Apr 6 at 20:02
1
$begingroup$
@MariaMazur For a much deeper understanding of so-called "Vieta jumping" see the literature I cite in my prior comment.
$endgroup$
– Bill Dubuque
Apr 6 at 20:04
$begingroup$
This one can be done by ID?
$endgroup$
– Maria Mazur
Apr 6 at 18:50
$begingroup$
This one can be done by ID?
$endgroup$
– Maria Mazur
Apr 6 at 18:50
1
1
$begingroup$
They say so here les-mathematiques.net/phorum/read.php?5,339463,339884 (in French)
$endgroup$
– Florian
Apr 6 at 18:51
$begingroup$
They say so here les-mathematiques.net/phorum/read.php?5,339463,339884 (in French)
$endgroup$
– Florian
Apr 6 at 18:51
1
1
$begingroup$
@DonThousand see zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf
$endgroup$
– Will Jagy
Apr 6 at 19:29
$begingroup$
@DonThousand see zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf
$endgroup$
– Will Jagy
Apr 6 at 19:29
2
2
$begingroup$
@DonThousand No, syymetries governing descent in the group of integer points on concis is much older - see the links I have here.
$endgroup$
– Bill Dubuque
Apr 6 at 20:02
$begingroup$
@DonThousand No, syymetries governing descent in the group of integer points on concis is much older - see the links I have here.
$endgroup$
– Bill Dubuque
Apr 6 at 20:02
1
1
$begingroup$
@MariaMazur For a much deeper understanding of so-called "Vieta jumping" see the literature I cite in my prior comment.
$endgroup$
– Bill Dubuque
Apr 6 at 20:04
$begingroup$
@MariaMazur For a much deeper understanding of so-called "Vieta jumping" see the literature I cite in my prior comment.
$endgroup$
– Bill Dubuque
Apr 6 at 20:04
|
show 5 more comments
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$begingroup$
I wonder why should this be closed?
$endgroup$
– Maria Mazur
Apr 6 at 20:30
2
$begingroup$
FWIW I voted to leave this open. However, I do have the misgiving that this type of a question does not have a "correct" answer. I guess opinions differ whether that makes a question unsuitable. It sounds like you welcome many answers, so may be using the "meta"tag big-list would be a good idea?
$endgroup$
– Jyrki Lahtonen
Apr 6 at 21:03