$existsinfty$ many pairs of consecutive squares s.t. their sum is also a square
$begingroup$
First of all, the term "pairs" is two of them, I assume (question's formulation is rather difficult to understand for me). So I guess this is the statement:
$$existsinftytext{ many pairs of consecutive squares }x^2, (x+1)^2text{ s.t. }x^2+(x+1)^2=y^2 x,yinmathbb{N}.$$
So there are two theorems I know of that might be of any use here:
1. An odd prime $p$ can be written as sum of squares iff $pequiv1$ mod $4$.
2. $ninmathbb{N}$ is the sum of squares iff primes that are $3$ mod $4$ occur an even number of times in the prime factorisation of $n$.
We can work out the LHS like so: $x^2+(x+1)^2=2x(x+1)+1$. We know that either $x$ or $x+1$ must be even; hence, the expression is of the form $4k+1$ with $kinmathbb{Z}$. From this we can conclude that the sum of two consecutive squares is always congruent $1$ modulo $4$ which also implies that primes that are $pequiv3$ mod $4$ that divide the sum of the consecutive squares must occur an even number of times. To this point, this does not prove anything significant, I think.
Does anyone have a hint on how to prove the statement?
number-theory modular-arithmetic sums-of-squares
asked Dec 16 '18 at 13:43
AlgebearAlgebear
655319
$endgroup$
add a comment |
$begingroup$
First of all, the term "pairs" is two of them, I assume (question's formulation is rather difficult to understand for me). So I guess this is the statement:
$$existsinftytext{ many pairs of consecutive squares }x^2, (x+1)^2text{ s.t. }x^2+(x+1)^2=y^2 x,yinmathbb{N}.$$
So there are two theorems I know of that might be of any use here:
1. An odd prime $p$ can be written as sum of squares iff $pequiv1$ mod $4$.
2. $ninmathbb{N}$ is the sum of squares iff primes that are $3$ mod $4$ occur an even number of times in the prime factorisation of $n$.
We can work out the LHS like so: $x^2+(x+1)^2=2x(x+1)+1$. We know that either $x$ or $x+1$ must be even; hence, the expression is of the form $4k+1$ with $kinmathbb{Z}$. From this we can conclude that the sum of two consecutive squares is always congruent $1$ modulo $4$ which also implies that primes that are $pequiv3$ mod $4$ that divide the sum of the consecutive squares must occur an even number of times. To this point, this does not prove anything significant, I think.
Does anyone have a hint on how to prove the statement?
number-theory modular-arithmetic sums-of-squares
asked Dec 16 '18 at 13:43
AlgebearAlgebear
655319
$endgroup$
3
$begingroup$
You want $$2x^2+2x+1=y^2iff 4x^2+4x+1+1=2y^2iff (2x+1)^2+1=2y^2$$ and this is a Pell equation of the form $z^2-2y^2=-1$.
$endgroup$
– Galc127
Dec 16 '18 at 14:02
3
$begingroup$
The list of values for $y$ is OEIS A$001653$. That link has multiple descriptions, including connections to Pell's Equation.
$endgroup$
– lulu
Dec 16 '18 at 14:03
add a comment |
$begingroup$
First of all, the term "pairs" is two of them, I assume (question's formulation is rather difficult to understand for me). So I guess this is the statement:
$$existsinftytext{ many pairs of consecutive squares }x^2, (x+1)^2text{ s.t. }x^2+(x+1)^2=y^2 x,yinmathbb{N}.$$
So there are two theorems I know of that might be of any use here:
1. An odd prime $p$ can be written as sum of squares iff $pequiv1$ mod $4$.
2. $ninmathbb{N}$ is the sum of squares iff primes that are $3$ mod $4$ occur an even number of times in the prime factorisation of $n$.
We can work out the LHS like so: $x^2+(x+1)^2=2x(x+1)+1$. We know that either $x$ or $x+1$ must be even; hence, the expression is of the form $4k+1$ with $kinmathbb{Z}$. From this we can conclude that the sum of two consecutive squares is always congruent $1$ modulo $4$ which also implies that primes that are $pequiv3$ mod $4$ that divide the sum of the consecutive squares must occur an even number of times. To this point, this does not prove anything significant, I think.
Does anyone have a hint on how to prove the statement?
number-theory modular-arithmetic sums-of-squares
asked Dec 16 '18 at 13:43
AlgebearAlgebear
655319
$endgroup$
First of all, the term "pairs" is two of them, I assume (question's formulation is rather difficult to understand for me). So I guess this is the statement:
$$existsinftytext{ many pairs of consecutive squares }x^2, (x+1)^2text{ s.t. }x^2+(x+1)^2=y^2 x,yinmathbb{N}.$$
So there are two theorems I know of that might be of any use here:
1. An odd prime $p$ can be written as sum of squares iff $pequiv1$ mod $4$.
2. $ninmathbb{N}$ is the sum of squares iff primes that are $3$ mod $4$ occur an even number of times in the prime factorisation of $n$.
We can work out the LHS like so: $x^2+(x+1)^2=2x(x+1)+1$. We know that either $x$ or $x+1$ must be even; hence, the expression is of the form $4k+1$ with $kinmathbb{Z}$. From this we can conclude that the sum of two consecutive squares is always congruent $1$ modulo $4$ which also implies that primes that are $pequiv3$ mod $4$ that divide the sum of the consecutive squares must occur an even number of times. To this point, this does not prove anything significant, I think.
Does anyone have a hint on how to prove the statement?
number-theory modular-arithmetic sums-of-squares
number-theory modular-arithmetic sums-of-squares
asked Dec 16 '18 at 13:43
AlgebearAlgebear
655319
asked Dec 16 '18 at 13:43
AlgebearAlgebear
655319
edited Dec 16 '18 at 13:55
Algebear
asked Dec 16 '18 at 13:43
AlgebearAlgebear
655319
asked Dec 16 '18 at 13:43
AlgebearAlgebear
655319
asked Dec 16 '18 at 13:43
AlgebearAlgebear
655319
655319
3
$begingroup$
You want $$2x^2+2x+1=y^2iff 4x^2+4x+1+1=2y^2iff (2x+1)^2+1=2y^2$$ and this is a Pell equation of the form $z^2-2y^2=-1$.
$endgroup$
– Galc127
Dec 16 '18 at 14:02
3
$begingroup$
The list of values for $y$ is OEIS A$001653$. That link has multiple descriptions, including connections to Pell's Equation.
$endgroup$
– lulu
Dec 16 '18 at 14:03
add a comment |
3
$begingroup$
You want $$2x^2+2x+1=y^2iff 4x^2+4x+1+1=2y^2iff (2x+1)^2+1=2y^2$$ and this is a Pell equation of the form $z^2-2y^2=-1$.
$endgroup$
– Galc127
Dec 16 '18 at 14:02
3
$begingroup$
The list of values for $y$ is OEIS A$001653$. That link has multiple descriptions, including connections to Pell's Equation.
$endgroup$
– lulu
Dec 16 '18 at 14:03
3
3
$begingroup$
You want $$2x^2+2x+1=y^2iff 4x^2+4x+1+1=2y^2iff (2x+1)^2+1=2y^2$$ and this is a Pell equation of the form $z^2-2y^2=-1$.
$endgroup$
– Galc127
Dec 16 '18 at 14:02
$begingroup$
You want $$2x^2+2x+1=y^2iff 4x^2+4x+1+1=2y^2iff (2x+1)^2+1=2y^2$$ and this is a Pell equation of the form $z^2-2y^2=-1$.
$endgroup$
– Galc127
Dec 16 '18 at 14:02
3
3
$begingroup$
The list of values for $y$ is OEIS A$001653$. That link has multiple descriptions, including connections to Pell's Equation.
$endgroup$
– lulu
Dec 16 '18 at 14:03
$begingroup$
The list of values for $y$ is OEIS A$001653$. That link has multiple descriptions, including connections to Pell's Equation.
$endgroup$
– lulu
Dec 16 '18 at 14:03
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3042615%2fexists-infty-many-pairs-of-consecutive-squares-s-t-their-sum-is-also-a-squar%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3042615%2fexists-infty-many-pairs-of-consecutive-squares-s-t-their-sum-is-also-a-squar%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
3
$begingroup$
You want $$2x^2+2x+1=y^2iff 4x^2+4x+1+1=2y^2iff (2x+1)^2+1=2y^2$$ and this is a Pell equation of the form $z^2-2y^2=-1$.
$endgroup$
– Galc127
Dec 16 '18 at 14:02
3
$begingroup$
The list of values for $y$ is OEIS A$001653$. That link has multiple descriptions, including connections to Pell's Equation.
$endgroup$
– lulu
Dec 16 '18 at 14:03