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Are there infinitely-many numbers that are both square and triangular?
Clash Royale CLAN TAG#URR8PPP
up vote
1
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I just started to read "Friendly Introduction to Number Theory" but I am getting stuck in the first exercises.
1.1. The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can
you figure out an efficient way to find triangular–square numbers? Do
you think that there are infinitely many?
https://www.math.brown.edu/~jhs/frintch1ch6.pdf
I found how to find out the number which is both square and triangle. (don't know if this is effective way)
https://github.com/y-zono/friendly-introduction-number-theory/blob/master/01/1-1/main.go
However how can I answer "Do you think that there are infinitely many?"? I think I need to find the formula but no idea yet. Can you please help me?
number-theory
edited 2 hours ago
Blue
45.1k868142
asked 2 hours ago
zono
1084
New contributor
zono is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
1
down vote
favorite
I just started to read "Friendly Introduction to Number Theory" but I am getting stuck in the first exercises.
1.1. The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can
you figure out an efficient way to find triangular–square numbers? Do
you think that there are infinitely many?
https://www.math.brown.edu/~jhs/frintch1ch6.pdf
I found how to find out the number which is both square and triangle. (don't know if this is effective way)
https://github.com/y-zono/friendly-introduction-number-theory/blob/master/01/1-1/main.go
However how can I answer "Do you think that there are infinitely many?"? I think I need to find the formula but no idea yet. Can you please help me?
number-theory
edited 2 hours ago
Blue
45.1k868142
asked 2 hours ago
zono
1084
New contributor
zono is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I just started to read "Friendly Introduction to Number Theory" but I am getting stuck in the first exercises.
1.1. The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can
you figure out an efficient way to find triangular–square numbers? Do
you think that there are infinitely many?
https://www.math.brown.edu/~jhs/frintch1ch6.pdf
I found how to find out the number which is both square and triangle. (don't know if this is effective way)
https://github.com/y-zono/friendly-introduction-number-theory/blob/master/01/1-1/main.go
However how can I answer "Do you think that there are infinitely many?"? I think I need to find the formula but no idea yet. Can you please help me?
number-theory
edited 2 hours ago
Blue
45.1k868142
asked 2 hours ago
zono
1084
New contributor
zono is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I just started to read "Friendly Introduction to Number Theory" but I am getting stuck in the first exercises.
1.1. The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can
you figure out an efficient way to find triangular–square numbers? Do
you think that there are infinitely many?
https://www.math.brown.edu/~jhs/frintch1ch6.pdf
I found how to find out the number which is both square and triangle. (don't know if this is effective way)
https://github.com/y-zono/friendly-introduction-number-theory/blob/master/01/1-1/main.go
However how can I answer "Do you think that there are infinitely many?"? I think I need to find the formula but no idea yet. Can you please help me?
number-theory
number-theory
edited 2 hours ago
Blue
45.1k868142
asked 2 hours ago
zono
1084
New contributor
zono is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Blue
45.1k868142
asked 2 hours ago
zono
1084
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zono is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Blue
45.1k868142
edited 2 hours ago
Blue
45.1k868142
edited 2 hours ago
Blue
45.1k868142
45.1k868142
asked 2 hours ago
zono
1084
New contributor
zono is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 2 hours ago
zono
1084
asked 2 hours ago
zono
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zono is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
zono is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |Â
4 Answers
4
active
oldest
votes
up vote
2
down vote
Have a look at https://en.wikipedia.org/wiki/Square_triangular_number
So, the answere is yes!
answered 2 hours ago
ALG
561
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ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you for great reference. I did not know the term "Square triangular number".
– zono
1 hour ago
add a comment |Â
up vote
1
down vote
The formula for triangle numbers is on page 9. Create an equation with the left being that formula using x, and the right the formula for squares, using y. X and y are not equal, but you can rearrange terms. Ask, when is the left divisible by something on the right? Are we sure that such divisibility exists for higher numbers?
Better, assume we have such an x and y. Can we always add something to each to get the next number?
answered 2 hours ago
Russ
478
Thank you! Now I'm trying to understand what you are saying... not easy for me. (my English skills is not good too)
– zono
1 hour ago
Good luck to you. The other answers are very detailed, but you have to do only enough to answer the question, are there infinitely many? I read the chapter, and it wants you to think about things that we can show go on forever, and other things that are conjectures (twin primes is one... we think it goes on forever but we do not know.)
– Russ
1 hour ago
Also, thank you for trying before asking, and welcome to the board! :-)
– Russ
1 hour ago
add a comment |Â
up vote
1
down vote
I think there are infinite such numbers, but they're rarer and rarer the farther the number from zero.
My computer gave these results:
1,
6, 35, 204, 1189, 6930., 40391, 235416. 1372105,
(The numbers which on squaring give triangular numbers until 10^7)
answered 2 hours ago
Harshith Vasireddy
111
New contributor
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you for answering. I agree that there are infinite such numbers. It took for a while on my PC to show the numbers.
– zono
2 hours ago
add a comment |Â
up vote
1
down vote
Of course the solution of equation:
$$Y^2=fracX(Xpm1)2$$
Defined solutions of Pell's equation: $$p^2-2s^2=pm1$$
But it is necessary to write the formula describing their solutions through solving Pell's equation:
$$X=p^2+4ps+4s^2$$
$$Y=p^2+3ps+2s^2$$
And more.
$$X=2s^2$$
$$Y=ps$$
$p,s$ - These numbers can be any character.
If you need to have a solution of the equation: $$Y^2=fracX(Xpma)2$$
It is necessary to substitute into the formulas uravneniyaPellya solutions: $$p^2-2s^2=pma$$
answered 2 hours ago
individ
3,2141715
Thank you so much! Give me for a while to understand it. So many information for me. I think it will take few days..
– zono
1 hour ago
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Have a look at https://en.wikipedia.org/wiki/Square_triangular_number
So, the answere is yes!
answered 2 hours ago
ALG
561
New contributor
ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you for great reference. I did not know the term "Square triangular number".
– zono
1 hour ago
add a comment |Â
up vote
2
down vote
Have a look at https://en.wikipedia.org/wiki/Square_triangular_number
So, the answere is yes!
answered 2 hours ago
ALG
561
New contributor
ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you for great reference. I did not know the term "Square triangular number".
– zono
1 hour ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Have a look at https://en.wikipedia.org/wiki/Square_triangular_number
So, the answere is yes!
answered 2 hours ago
ALG
561
New contributor
ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Have a look at https://en.wikipedia.org/wiki/Square_triangular_number
So, the answere is yes!
answered 2 hours ago
ALG
561
New contributor
ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 hours ago
ALG
561
New contributor
ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 hours ago
ALG
561
answered 2 hours ago
ALG
561
561
New contributor
ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
ALG is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you for great reference. I did not know the term "Square triangular number".
– zono
1 hour ago
add a comment |Â
Thank you for great reference. I did not know the term "Square triangular number".
– zono
1 hour ago
Thank you for great reference. I did not know the term "Square triangular number".
– zono
1 hour ago
Thank you for great reference. I did not know the term "Square triangular number".
– zono
1 hour ago
add a comment |Â
up vote
1
down vote
The formula for triangle numbers is on page 9. Create an equation with the left being that formula using x, and the right the formula for squares, using y. X and y are not equal, but you can rearrange terms. Ask, when is the left divisible by something on the right? Are we sure that such divisibility exists for higher numbers?
Better, assume we have such an x and y. Can we always add something to each to get the next number?
answered 2 hours ago
Russ
478
Thank you! Now I'm trying to understand what you are saying... not easy for me. (my English skills is not good too)
– zono
1 hour ago
Good luck to you. The other answers are very detailed, but you have to do only enough to answer the question, are there infinitely many? I read the chapter, and it wants you to think about things that we can show go on forever, and other things that are conjectures (twin primes is one... we think it goes on forever but we do not know.)
– Russ
1 hour ago
Also, thank you for trying before asking, and welcome to the board! :-)
– Russ
1 hour ago
add a comment |Â
up vote
1
down vote
The formula for triangle numbers is on page 9. Create an equation with the left being that formula using x, and the right the formula for squares, using y. X and y are not equal, but you can rearrange terms. Ask, when is the left divisible by something on the right? Are we sure that such divisibility exists for higher numbers?
Better, assume we have such an x and y. Can we always add something to each to get the next number?
answered 2 hours ago
Russ
478
Thank you! Now I'm trying to understand what you are saying... not easy for me. (my English skills is not good too)
– zono
1 hour ago
Good luck to you. The other answers are very detailed, but you have to do only enough to answer the question, are there infinitely many? I read the chapter, and it wants you to think about things that we can show go on forever, and other things that are conjectures (twin primes is one... we think it goes on forever but we do not know.)
– Russ
1 hour ago
Also, thank you for trying before asking, and welcome to the board! :-)
– Russ
1 hour ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The formula for triangle numbers is on page 9. Create an equation with the left being that formula using x, and the right the formula for squares, using y. X and y are not equal, but you can rearrange terms. Ask, when is the left divisible by something on the right? Are we sure that such divisibility exists for higher numbers?
Better, assume we have such an x and y. Can we always add something to each to get the next number?
answered 2 hours ago
Russ
478
The formula for triangle numbers is on page 9. Create an equation with the left being that formula using x, and the right the formula for squares, using y. X and y are not equal, but you can rearrange terms. Ask, when is the left divisible by something on the right? Are we sure that such divisibility exists for higher numbers?
Better, assume we have such an x and y. Can we always add something to each to get the next number?
answered 2 hours ago
Russ
478
answered 2 hours ago
Russ
478
answered 2 hours ago
Russ
478
answered 2 hours ago
Russ
478
478
Thank you! Now I'm trying to understand what you are saying... not easy for me. (my English skills is not good too)
– zono
1 hour ago
Good luck to you. The other answers are very detailed, but you have to do only enough to answer the question, are there infinitely many? I read the chapter, and it wants you to think about things that we can show go on forever, and other things that are conjectures (twin primes is one... we think it goes on forever but we do not know.)
– Russ
1 hour ago
Also, thank you for trying before asking, and welcome to the board! :-)
– Russ
1 hour ago
add a comment |Â
Thank you! Now I'm trying to understand what you are saying... not easy for me. (my English skills is not good too)
– zono
1 hour ago
Good luck to you. The other answers are very detailed, but you have to do only enough to answer the question, are there infinitely many? I read the chapter, and it wants you to think about things that we can show go on forever, and other things that are conjectures (twin primes is one... we think it goes on forever but we do not know.)
– Russ
1 hour ago
Also, thank you for trying before asking, and welcome to the board! :-)
– Russ
1 hour ago
Thank you! Now I'm trying to understand what you are saying... not easy for me. (my English skills is not good too)
– zono
1 hour ago
Thank you! Now I'm trying to understand what you are saying... not easy for me. (my English skills is not good too)
– zono
1 hour ago
Good luck to you. The other answers are very detailed, but you have to do only enough to answer the question, are there infinitely many? I read the chapter, and it wants you to think about things that we can show go on forever, and other things that are conjectures (twin primes is one... we think it goes on forever but we do not know.)
– Russ
1 hour ago
Good luck to you. The other answers are very detailed, but you have to do only enough to answer the question, are there infinitely many? I read the chapter, and it wants you to think about things that we can show go on forever, and other things that are conjectures (twin primes is one... we think it goes on forever but we do not know.)
– Russ
1 hour ago
Also, thank you for trying before asking, and welcome to the board! :-)
– Russ
1 hour ago
Also, thank you for trying before asking, and welcome to the board! :-)
– Russ
1 hour ago
add a comment |Â
up vote
1
down vote
I think there are infinite such numbers, but they're rarer and rarer the farther the number from zero.
My computer gave these results:
1,
6, 35, 204, 1189, 6930., 40391, 235416. 1372105,
(The numbers which on squaring give triangular numbers until 10^7)
answered 2 hours ago
Harshith Vasireddy
111
New contributor
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you for answering. I agree that there are infinite such numbers. It took for a while on my PC to show the numbers.
– zono
2 hours ago
add a comment |Â
up vote
1
down vote
I think there are infinite such numbers, but they're rarer and rarer the farther the number from zero.
My computer gave these results:
1,
6, 35, 204, 1189, 6930., 40391, 235416. 1372105,
(The numbers which on squaring give triangular numbers until 10^7)
answered 2 hours ago
Harshith Vasireddy
111
New contributor
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you for answering. I agree that there are infinite such numbers. It took for a while on my PC to show the numbers.
– zono
2 hours ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
I think there are infinite such numbers, but they're rarer and rarer the farther the number from zero.
My computer gave these results:
1,
6, 35, 204, 1189, 6930., 40391, 235416. 1372105,
(The numbers which on squaring give triangular numbers until 10^7)
answered 2 hours ago
Harshith Vasireddy
111
New contributor
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I think there are infinite such numbers, but they're rarer and rarer the farther the number from zero.
My computer gave these results:
1,
6, 35, 204, 1189, 6930., 40391, 235416. 1372105,
(The numbers which on squaring give triangular numbers until 10^7)
answered 2 hours ago
Harshith Vasireddy
111
New contributor
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 hours ago
Harshith Vasireddy
111
New contributor
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 hours ago
Harshith Vasireddy
111
answered 2 hours ago
Harshith Vasireddy
111
111
New contributor
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Harshith Vasireddy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you for answering. I agree that there are infinite such numbers. It took for a while on my PC to show the numbers.
– zono
2 hours ago
add a comment |Â
Thank you for answering. I agree that there are infinite such numbers. It took for a while on my PC to show the numbers.
– zono
2 hours ago
Thank you for answering. I agree that there are infinite such numbers. It took for a while on my PC to show the numbers.
– zono
2 hours ago
Thank you for answering. I agree that there are infinite such numbers. It took for a while on my PC to show the numbers.
– zono
2 hours ago
add a comment |Â
up vote
1
down vote
Of course the solution of equation:
$$Y^2=fracX(Xpm1)2$$
Defined solutions of Pell's equation: $$p^2-2s^2=pm1$$
But it is necessary to write the formula describing their solutions through solving Pell's equation:
$$X=p^2+4ps+4s^2$$
$$Y=p^2+3ps+2s^2$$
And more.
$$X=2s^2$$
$$Y=ps$$
$p,s$ - These numbers can be any character.
If you need to have a solution of the equation: $$Y^2=fracX(Xpma)2$$
It is necessary to substitute into the formulas uravneniyaPellya solutions: $$p^2-2s^2=pma$$
answered 2 hours ago
individ
3,2141715
Thank you so much! Give me for a while to understand it. So many information for me. I think it will take few days..
– zono
1 hour ago
add a comment |Â
up vote
1
down vote
Of course the solution of equation:
$$Y^2=fracX(Xpm1)2$$
Defined solutions of Pell's equation: $$p^2-2s^2=pm1$$
But it is necessary to write the formula describing their solutions through solving Pell's equation:
$$X=p^2+4ps+4s^2$$
$$Y=p^2+3ps+2s^2$$
And more.
$$X=2s^2$$
$$Y=ps$$
$p,s$ - These numbers can be any character.
If you need to have a solution of the equation: $$Y^2=fracX(Xpma)2$$
It is necessary to substitute into the formulas uravneniyaPellya solutions: $$p^2-2s^2=pma$$
answered 2 hours ago
individ
3,2141715
Thank you so much! Give me for a while to understand it. So many information for me. I think it will take few days..
– zono
1 hour ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Of course the solution of equation:
$$Y^2=fracX(Xpm1)2$$
Defined solutions of Pell's equation: $$p^2-2s^2=pm1$$
But it is necessary to write the formula describing their solutions through solving Pell's equation:
$$X=p^2+4ps+4s^2$$
$$Y=p^2+3ps+2s^2$$
And more.
$$X=2s^2$$
$$Y=ps$$
$p,s$ - These numbers can be any character.
If you need to have a solution of the equation: $$Y^2=fracX(Xpma)2$$
It is necessary to substitute into the formulas uravneniyaPellya solutions: $$p^2-2s^2=pma$$
answered 2 hours ago
individ
3,2141715
Of course the solution of equation:
$$Y^2=fracX(Xpm1)2$$
Defined solutions of Pell's equation: $$p^2-2s^2=pm1$$
But it is necessary to write the formula describing their solutions through solving Pell's equation:
$$X=p^2+4ps+4s^2$$
$$Y=p^2+3ps+2s^2$$
And more.
$$X=2s^2$$
$$Y=ps$$
$p,s$ - These numbers can be any character.
If you need to have a solution of the equation: $$Y^2=fracX(Xpma)2$$
It is necessary to substitute into the formulas uravneniyaPellya solutions: $$p^2-2s^2=pma$$
answered 2 hours ago
individ
3,2141715
answered 2 hours ago
individ
3,2141715
answered 2 hours ago
individ
3,2141715
answered 2 hours ago
individ
3,2141715
3,2141715
Thank you so much! Give me for a while to understand it. So many information for me. I think it will take few days..
– zono
1 hour ago
add a comment |Â
Thank you so much! Give me for a while to understand it. So many information for me. I think it will take few days..
– zono
1 hour ago
Thank you so much! Give me for a while to understand it. So many information for me. I think it will take few days..
– zono
1 hour ago
Thank you so much! Give me for a while to understand it. So many information for me. I think it will take few days..
– zono
1 hour ago
add a comment |Â
zono is a new contributor. Be nice, and check out our Code of Conduct.
zono is a new contributor. Be nice, and check out our Code of Conduct.
zono is a new contributor. Be nice, and check out our Code of Conduct.
zono is a new contributor. Be nice, and check out our Code of Conduct.
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