How do I find all prime solutions $p, q, r$ of the equation $displaystyle p(p+1)+q(q+1) = r(r+1)$?
Find primes $p, q, r$ of the equation $$p(p+1)+q(q+1)
= r(r+1)$$
I know that it has only one solution namely $p = q = 2,r = 3$. But i can't show that.
Thank you for any help
number-theory elementary-number-theory prime-numbers divisibility
add a comment |
Find primes $p, q, r$ of the equation $$p(p+1)+q(q+1)
= r(r+1)$$
I know that it has only one solution namely $p = q = 2,r = 3$. But i can't show that.
Thank you for any help
number-theory elementary-number-theory prime-numbers divisibility
How do you know that no other solutions exist then, what is your source ?
– Peter
Dec 12 '16 at 22:28
I think this problem related to A shinzel solution of the titled equation if i'm true
– zeraoulia rafik
Dec 12 '16 at 22:43
It may or may not help to rephrase it as $p^2 + p + q^2 + q = r^2 + r$.
– Robert Soupe
Dec 13 '16 at 2:21
this equivalent to solve $$(2p+1)^2+(2q+1)^2=(2r+1)^2+1$$
– Bumblebee
Jan 18 '18 at 20:01
add a comment |
Find primes $p, q, r$ of the equation $$p(p+1)+q(q+1)
= r(r+1)$$
I know that it has only one solution namely $p = q = 2,r = 3$. But i can't show that.
Thank you for any help
number-theory elementary-number-theory prime-numbers divisibility
Find primes $p, q, r$ of the equation $$p(p+1)+q(q+1)
= r(r+1)$$
I know that it has only one solution namely $p = q = 2,r = 3$. But i can't show that.
Thank you for any help
number-theory elementary-number-theory prime-numbers divisibility
number-theory elementary-number-theory prime-numbers divisibility
edited Jan 3 at 13:50
greedoid
38.5k114797
38.5k114797
asked Dec 12 '16 at 22:12
Youssra El Yossra YoussraYoussra El Yossra Youssra
975
975
How do you know that no other solutions exist then, what is your source ?
– Peter
Dec 12 '16 at 22:28
I think this problem related to A shinzel solution of the titled equation if i'm true
– zeraoulia rafik
Dec 12 '16 at 22:43
It may or may not help to rephrase it as $p^2 + p + q^2 + q = r^2 + r$.
– Robert Soupe
Dec 13 '16 at 2:21
this equivalent to solve $$(2p+1)^2+(2q+1)^2=(2r+1)^2+1$$
– Bumblebee
Jan 18 '18 at 20:01
add a comment |
How do you know that no other solutions exist then, what is your source ?
– Peter
Dec 12 '16 at 22:28
I think this problem related to A shinzel solution of the titled equation if i'm true
– zeraoulia rafik
Dec 12 '16 at 22:43
It may or may not help to rephrase it as $p^2 + p + q^2 + q = r^2 + r$.
– Robert Soupe
Dec 13 '16 at 2:21
this equivalent to solve $$(2p+1)^2+(2q+1)^2=(2r+1)^2+1$$
– Bumblebee
Jan 18 '18 at 20:01
How do you know that no other solutions exist then, what is your source ?
– Peter
Dec 12 '16 at 22:28
How do you know that no other solutions exist then, what is your source ?
– Peter
Dec 12 '16 at 22:28
I think this problem related to A shinzel solution of the titled equation if i'm true
– zeraoulia rafik
Dec 12 '16 at 22:43
I think this problem related to A shinzel solution of the titled equation if i'm true
– zeraoulia rafik
Dec 12 '16 at 22:43
It may or may not help to rephrase it as $p^2 + p + q^2 + q = r^2 + r$.
– Robert Soupe
Dec 13 '16 at 2:21
It may or may not help to rephrase it as $p^2 + p + q^2 + q = r^2 + r$.
– Robert Soupe
Dec 13 '16 at 2:21
this equivalent to solve $$(2p+1)^2+(2q+1)^2=(2r+1)^2+1$$
– Bumblebee
Jan 18 '18 at 20:01
this equivalent to solve $$(2p+1)^2+(2q+1)^2=(2r+1)^2+1$$
– Bumblebee
Jan 18 '18 at 20:01
add a comment |
3 Answers
3
active
oldest
votes
May this lead to a simple proof for your problem according to your unic example of solution . There is only one
solution, namely $p = q = 2,r = 3$. To see that, we shall find all solutions
of the equation $p(p+1)+q(q+1) = n(n+1)$ where $p$ and $q$ are primes and
$n$ is a positive integer. Our equation yields
$p(p+1) = n(n+1)-q(q+1) = (n-q)(n+q+1)$,
and we must have $n > q$. Since $p$ is a prime, we have either $p|n-q$ or
$p|n+q+1$. If $p|n-q$, then we have $pleq n-q$, which implies $p(p+1)
leq (n-q)(n-q+1)$, and therefore $n+q+1 leq n-q+1$, which is impossible.
Thus we have $p|n+q+ 1$, which means that for some positive integer $k$
,$n+q+1 = kp$, which implies $p+1 = k(n-q)tag1$.
If we had $k = 1$, then $n+q+ 1 = p$ and $p+ 1 = n-q$, which gives $p-q
= n+ 1$ and $p+q = n- 1$, which is impossible, because $(p+q)-(p-q)=2q>0$ and $(n-1)-(n+1)=-2<0$. Thus, $k > 1$. From $(1)$ we
easily obtain:
$$begin{align}
2q &= (n+q)-(n-q) \
&= kp-1-(n-q) \
&= k[k(n-q)-1]-1-(n-q) \
&= (k+1)[(k-1)(n-q)-1].
end{align}$$
Since $k geq 2$, we have $k+1 geq 3$. The last equality, whose left-hand side has
positive integer divisors $1, 2, q$, and $2q$ only, implies that either $k+ 1 = q$
or $k+1 = 2q$. If $k+1 = q$, then $(k-1)(n-q) = 3$, hence $(q-2)(n-q) = 3$.
This leads to either $q-2 = 1$, $n-q = 3$, that is $q = 3, n = 6, k = q-1 = 2$,
and, in view of $(1)$, $p = 5$, or else, $q-2 = 3$, $n-q = 1$, which gives $q = 5,
n = 6, k = 4$, and in view of $(1)$, $p = 3$.
On the other hand, if $k+1 = 2q$, then $(k-1)(n-q) = 2$, hence
$2(q-1)(n-q) = 2$. This leads to $q-1 = 1$ and $n-q = 1$, or $q = 2, n = 3$,
and, in view of $(1)$, $p = 2$. Thus, for positive integer $ n$, we have the
following solutions in primes $p$ and $q$:
$(p = q = 2, n = 3; 2)$,
$ (p = 5,
q = 3, n = 6)$, and$(p = 3, q = 5, n = 6)$.
Only in the first solution all
three numbers are primes.
Note: If we denote by $displaystyle t_n = frac{n(n+1)}{2}$ the nth triangular number,
then the equation $t_p+t_q = t_r$
has only one solution in prime numbers, namely $p = q = 2, r = 3$.
It should be $p+q=n-1$.
– user236182
Dec 12 '16 at 22:51
It's still impossible because $(p+q)-(p-q)=2q>0$ while $(n-1)-(n+1)=-2<0$.
– user236182
Dec 12 '16 at 22:54
for k=1 it's impossible and i cited your case p+q=n-1 in the side of n+q=p-1
– zeraoulia rafik
Dec 12 '16 at 22:55
If $p+1=n-q$, then $p+q=n-1$.
– user236182
Dec 12 '16 at 22:57
And if $n+q+1=p$, then $p-q=n+1$.
– user236182
Dec 12 '16 at 22:58
|
show 3 more comments
Case $pne q$: We can assume that $p>q$, then $$r^2+r leq (p-1)^2+p^2+p-1+p=2p^2implies boxed{rleq psqrt{2}}$$
From:
$$p(p+1) =r^2- q^2+r-q =(r-q)(r+q+1)$$
we get
If $pmid r-q$ then $r+q+1mid p+1$, so $pleq r-q<r$ and $r+q+1leq p+1 implies r<p$ a contradiction.
If $pmid r+q+1$ then $r-qmid p+1$. Since $r+q+1=kp$ for some integer $kgeq 1$ we have $$kpleq psqrt{2}+(p-1)+1 implies kleq 2$$
Case 1: $k=2$ we get $r+q+1=2p$ and $p+1=2r-2q$ from where we get $3p=11$, no good.
Case 2: $k=1$ we get $r+q+1=p$ and $p+1=r-q$ so $q=-1$ again contradiction.
So $p=q$ and now we have to solve $$2p^2+2p = r^2+r$$
So $$2p(p+1)=r(r+1)implies rmid 2p(p+1)$$
If $rmid 2$ then $r=2$ which is impossibile.
If $rmid p$ then $rleq p$ which is impossibile.
If $rmid p+1$ then $rleq p+1$ but then $r=p+1$ since $r>p$, so $r=3$ and $p=2$.
add a comment |
From the given equation
$$p(p+1)+q(q+1)=r(r+1)$$
it follows that $p < r$ and $q < r$.
Next, another inequality which will be useful later . . .
Claim:$;p+q > r$.
Proof:
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(p^2+q^2)+(p+q)=r(r+1)\[4pt]
implies;&(p+q)^2+(p+q)>r(r+1)\[4pt]
implies;&(p+q)(p+q+1)>r(r+1)\[4pt]
implies;&p+q>r\[4pt]
end{align*}
as claimed.
Returning to the main problem . . .
First suppose $p=q$.
Then the given equation reduces to
$$2p(p+1)=r(r+1)$$
hence, since $r > p$, it follows that $r|(p+1)$.
But then $p < r le p+1$, so $r=p+1$, which implies $p=2$, and $r=3$.
It can be verified that the triple $(p,q,r)=(2,2,3)$ satisfies the given equation.
Next suppose $p,q$ are distinct.
Without loss of generality, assume $p < q$.
Suppose $;p=2$.
Then from $p < q < r$, we get $qge 3$ and $rge 5$, hence
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(2)(2+1)+q(q+1)=r(r+1)\[4pt]
implies;&r(r+1)-q(q+1)=6\[4pt]
implies;&(r-q)(q+r+1)=6\[4pt]
implies;&(q+r+1)mid 6\[4pt]
implies;&q+r+1le 6\[4pt]
end{align*}
contradiction, since $q+r+1ge 3+5+1=9$.
Hence we must have $p > 2$.
Since $p+q > r$, it follows that $pnotmid r-q$, and $qnotmid r-p$.
begin{align*}
text{Then};;&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&p(p+1)=r(r+1)-q(q+1)\[4pt]
implies;&p(p+1)=(q+r+1)(r-q)\[4pt]
implies;&pmid (q+r+1)\[4pt]
implies;&pmid (p+q+r+1)\[12pt]
text{and};,&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&q(q+1)=r(r+1)-p(p+1)\[4pt]
implies;&q(q+1)=(p+r+1)(r-p)\[4pt]
implies;&qmid (p+r+1)\[4pt]
implies;&qmid (p+q+r+1)\[12pt]
text{hence};,&pqmid (p+q+r+1)\[4pt]
implies;&pqle p+q+r+1\[4pt]
implies;&pq < p+q+(p+q)+1\[4pt]
implies;&pq-2p-2q < 1\[4pt]
implies;&(p-2)(q-2) < 5\[4pt]
implies;&q-2 < 5\[4pt]
implies;&q < 7\[4pt]
implies;&qle 5\[4pt]
implies;&(p,q)=(3,5)\[4pt]
implies;&(3)(3+1)+(5)(5+1)=r(r+1)\[4pt]
implies;&r=6\[4pt]
end{align*}
contradiction, since $6$ is not prime.
Therefore the only solution is $(p,q,r)=(2,2,3)$.
add a comment |
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%2f2056120%2fhow-do-i-find-all-prime-solutions-p-q-r-of-the-equation-displaystyle-pp1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
May this lead to a simple proof for your problem according to your unic example of solution . There is only one
solution, namely $p = q = 2,r = 3$. To see that, we shall find all solutions
of the equation $p(p+1)+q(q+1) = n(n+1)$ where $p$ and $q$ are primes and
$n$ is a positive integer. Our equation yields
$p(p+1) = n(n+1)-q(q+1) = (n-q)(n+q+1)$,
and we must have $n > q$. Since $p$ is a prime, we have either $p|n-q$ or
$p|n+q+1$. If $p|n-q$, then we have $pleq n-q$, which implies $p(p+1)
leq (n-q)(n-q+1)$, and therefore $n+q+1 leq n-q+1$, which is impossible.
Thus we have $p|n+q+ 1$, which means that for some positive integer $k$
,$n+q+1 = kp$, which implies $p+1 = k(n-q)tag1$.
If we had $k = 1$, then $n+q+ 1 = p$ and $p+ 1 = n-q$, which gives $p-q
= n+ 1$ and $p+q = n- 1$, which is impossible, because $(p+q)-(p-q)=2q>0$ and $(n-1)-(n+1)=-2<0$. Thus, $k > 1$. From $(1)$ we
easily obtain:
$$begin{align}
2q &= (n+q)-(n-q) \
&= kp-1-(n-q) \
&= k[k(n-q)-1]-1-(n-q) \
&= (k+1)[(k-1)(n-q)-1].
end{align}$$
Since $k geq 2$, we have $k+1 geq 3$. The last equality, whose left-hand side has
positive integer divisors $1, 2, q$, and $2q$ only, implies that either $k+ 1 = q$
or $k+1 = 2q$. If $k+1 = q$, then $(k-1)(n-q) = 3$, hence $(q-2)(n-q) = 3$.
This leads to either $q-2 = 1$, $n-q = 3$, that is $q = 3, n = 6, k = q-1 = 2$,
and, in view of $(1)$, $p = 5$, or else, $q-2 = 3$, $n-q = 1$, which gives $q = 5,
n = 6, k = 4$, and in view of $(1)$, $p = 3$.
On the other hand, if $k+1 = 2q$, then $(k-1)(n-q) = 2$, hence
$2(q-1)(n-q) = 2$. This leads to $q-1 = 1$ and $n-q = 1$, or $q = 2, n = 3$,
and, in view of $(1)$, $p = 2$. Thus, for positive integer $ n$, we have the
following solutions in primes $p$ and $q$:
$(p = q = 2, n = 3; 2)$,
$ (p = 5,
q = 3, n = 6)$, and$(p = 3, q = 5, n = 6)$.
Only in the first solution all
three numbers are primes.
Note: If we denote by $displaystyle t_n = frac{n(n+1)}{2}$ the nth triangular number,
then the equation $t_p+t_q = t_r$
has only one solution in prime numbers, namely $p = q = 2, r = 3$.
It should be $p+q=n-1$.
– user236182
Dec 12 '16 at 22:51
It's still impossible because $(p+q)-(p-q)=2q>0$ while $(n-1)-(n+1)=-2<0$.
– user236182
Dec 12 '16 at 22:54
for k=1 it's impossible and i cited your case p+q=n-1 in the side of n+q=p-1
– zeraoulia rafik
Dec 12 '16 at 22:55
If $p+1=n-q$, then $p+q=n-1$.
– user236182
Dec 12 '16 at 22:57
And if $n+q+1=p$, then $p-q=n+1$.
– user236182
Dec 12 '16 at 22:58
|
show 3 more comments
May this lead to a simple proof for your problem according to your unic example of solution . There is only one
solution, namely $p = q = 2,r = 3$. To see that, we shall find all solutions
of the equation $p(p+1)+q(q+1) = n(n+1)$ where $p$ and $q$ are primes and
$n$ is a positive integer. Our equation yields
$p(p+1) = n(n+1)-q(q+1) = (n-q)(n+q+1)$,
and we must have $n > q$. Since $p$ is a prime, we have either $p|n-q$ or
$p|n+q+1$. If $p|n-q$, then we have $pleq n-q$, which implies $p(p+1)
leq (n-q)(n-q+1)$, and therefore $n+q+1 leq n-q+1$, which is impossible.
Thus we have $p|n+q+ 1$, which means that for some positive integer $k$
,$n+q+1 = kp$, which implies $p+1 = k(n-q)tag1$.
If we had $k = 1$, then $n+q+ 1 = p$ and $p+ 1 = n-q$, which gives $p-q
= n+ 1$ and $p+q = n- 1$, which is impossible, because $(p+q)-(p-q)=2q>0$ and $(n-1)-(n+1)=-2<0$. Thus, $k > 1$. From $(1)$ we
easily obtain:
$$begin{align}
2q &= (n+q)-(n-q) \
&= kp-1-(n-q) \
&= k[k(n-q)-1]-1-(n-q) \
&= (k+1)[(k-1)(n-q)-1].
end{align}$$
Since $k geq 2$, we have $k+1 geq 3$. The last equality, whose left-hand side has
positive integer divisors $1, 2, q$, and $2q$ only, implies that either $k+ 1 = q$
or $k+1 = 2q$. If $k+1 = q$, then $(k-1)(n-q) = 3$, hence $(q-2)(n-q) = 3$.
This leads to either $q-2 = 1$, $n-q = 3$, that is $q = 3, n = 6, k = q-1 = 2$,
and, in view of $(1)$, $p = 5$, or else, $q-2 = 3$, $n-q = 1$, which gives $q = 5,
n = 6, k = 4$, and in view of $(1)$, $p = 3$.
On the other hand, if $k+1 = 2q$, then $(k-1)(n-q) = 2$, hence
$2(q-1)(n-q) = 2$. This leads to $q-1 = 1$ and $n-q = 1$, or $q = 2, n = 3$,
and, in view of $(1)$, $p = 2$. Thus, for positive integer $ n$, we have the
following solutions in primes $p$ and $q$:
$(p = q = 2, n = 3; 2)$,
$ (p = 5,
q = 3, n = 6)$, and$(p = 3, q = 5, n = 6)$.
Only in the first solution all
three numbers are primes.
Note: If we denote by $displaystyle t_n = frac{n(n+1)}{2}$ the nth triangular number,
then the equation $t_p+t_q = t_r$
has only one solution in prime numbers, namely $p = q = 2, r = 3$.
It should be $p+q=n-1$.
– user236182
Dec 12 '16 at 22:51
It's still impossible because $(p+q)-(p-q)=2q>0$ while $(n-1)-(n+1)=-2<0$.
– user236182
Dec 12 '16 at 22:54
for k=1 it's impossible and i cited your case p+q=n-1 in the side of n+q=p-1
– zeraoulia rafik
Dec 12 '16 at 22:55
If $p+1=n-q$, then $p+q=n-1$.
– user236182
Dec 12 '16 at 22:57
And if $n+q+1=p$, then $p-q=n+1$.
– user236182
Dec 12 '16 at 22:58
|
show 3 more comments
May this lead to a simple proof for your problem according to your unic example of solution . There is only one
solution, namely $p = q = 2,r = 3$. To see that, we shall find all solutions
of the equation $p(p+1)+q(q+1) = n(n+1)$ where $p$ and $q$ are primes and
$n$ is a positive integer. Our equation yields
$p(p+1) = n(n+1)-q(q+1) = (n-q)(n+q+1)$,
and we must have $n > q$. Since $p$ is a prime, we have either $p|n-q$ or
$p|n+q+1$. If $p|n-q$, then we have $pleq n-q$, which implies $p(p+1)
leq (n-q)(n-q+1)$, and therefore $n+q+1 leq n-q+1$, which is impossible.
Thus we have $p|n+q+ 1$, which means that for some positive integer $k$
,$n+q+1 = kp$, which implies $p+1 = k(n-q)tag1$.
If we had $k = 1$, then $n+q+ 1 = p$ and $p+ 1 = n-q$, which gives $p-q
= n+ 1$ and $p+q = n- 1$, which is impossible, because $(p+q)-(p-q)=2q>0$ and $(n-1)-(n+1)=-2<0$. Thus, $k > 1$. From $(1)$ we
easily obtain:
$$begin{align}
2q &= (n+q)-(n-q) \
&= kp-1-(n-q) \
&= k[k(n-q)-1]-1-(n-q) \
&= (k+1)[(k-1)(n-q)-1].
end{align}$$
Since $k geq 2$, we have $k+1 geq 3$. The last equality, whose left-hand side has
positive integer divisors $1, 2, q$, and $2q$ only, implies that either $k+ 1 = q$
or $k+1 = 2q$. If $k+1 = q$, then $(k-1)(n-q) = 3$, hence $(q-2)(n-q) = 3$.
This leads to either $q-2 = 1$, $n-q = 3$, that is $q = 3, n = 6, k = q-1 = 2$,
and, in view of $(1)$, $p = 5$, or else, $q-2 = 3$, $n-q = 1$, which gives $q = 5,
n = 6, k = 4$, and in view of $(1)$, $p = 3$.
On the other hand, if $k+1 = 2q$, then $(k-1)(n-q) = 2$, hence
$2(q-1)(n-q) = 2$. This leads to $q-1 = 1$ and $n-q = 1$, or $q = 2, n = 3$,
and, in view of $(1)$, $p = 2$. Thus, for positive integer $ n$, we have the
following solutions in primes $p$ and $q$:
$(p = q = 2, n = 3; 2)$,
$ (p = 5,
q = 3, n = 6)$, and$(p = 3, q = 5, n = 6)$.
Only in the first solution all
three numbers are primes.
Note: If we denote by $displaystyle t_n = frac{n(n+1)}{2}$ the nth triangular number,
then the equation $t_p+t_q = t_r$
has only one solution in prime numbers, namely $p = q = 2, r = 3$.
May this lead to a simple proof for your problem according to your unic example of solution . There is only one
solution, namely $p = q = 2,r = 3$. To see that, we shall find all solutions
of the equation $p(p+1)+q(q+1) = n(n+1)$ where $p$ and $q$ are primes and
$n$ is a positive integer. Our equation yields
$p(p+1) = n(n+1)-q(q+1) = (n-q)(n+q+1)$,
and we must have $n > q$. Since $p$ is a prime, we have either $p|n-q$ or
$p|n+q+1$. If $p|n-q$, then we have $pleq n-q$, which implies $p(p+1)
leq (n-q)(n-q+1)$, and therefore $n+q+1 leq n-q+1$, which is impossible.
Thus we have $p|n+q+ 1$, which means that for some positive integer $k$
,$n+q+1 = kp$, which implies $p+1 = k(n-q)tag1$.
If we had $k = 1$, then $n+q+ 1 = p$ and $p+ 1 = n-q$, which gives $p-q
= n+ 1$ and $p+q = n- 1$, which is impossible, because $(p+q)-(p-q)=2q>0$ and $(n-1)-(n+1)=-2<0$. Thus, $k > 1$. From $(1)$ we
easily obtain:
$$begin{align}
2q &= (n+q)-(n-q) \
&= kp-1-(n-q) \
&= k[k(n-q)-1]-1-(n-q) \
&= (k+1)[(k-1)(n-q)-1].
end{align}$$
Since $k geq 2$, we have $k+1 geq 3$. The last equality, whose left-hand side has
positive integer divisors $1, 2, q$, and $2q$ only, implies that either $k+ 1 = q$
or $k+1 = 2q$. If $k+1 = q$, then $(k-1)(n-q) = 3$, hence $(q-2)(n-q) = 3$.
This leads to either $q-2 = 1$, $n-q = 3$, that is $q = 3, n = 6, k = q-1 = 2$,
and, in view of $(1)$, $p = 5$, or else, $q-2 = 3$, $n-q = 1$, which gives $q = 5,
n = 6, k = 4$, and in view of $(1)$, $p = 3$.
On the other hand, if $k+1 = 2q$, then $(k-1)(n-q) = 2$, hence
$2(q-1)(n-q) = 2$. This leads to $q-1 = 1$ and $n-q = 1$, or $q = 2, n = 3$,
and, in view of $(1)$, $p = 2$. Thus, for positive integer $ n$, we have the
following solutions in primes $p$ and $q$:
$(p = q = 2, n = 3; 2)$,
$ (p = 5,
q = 3, n = 6)$, and$(p = 3, q = 5, n = 6)$.
Only in the first solution all
three numbers are primes.
Note: If we denote by $displaystyle t_n = frac{n(n+1)}{2}$ the nth triangular number,
then the equation $t_p+t_q = t_r$
has only one solution in prime numbers, namely $p = q = 2, r = 3$.
edited Jan 4 at 12:55
Shaun
8,820113681
8,820113681
answered Dec 12 '16 at 22:46
zeraoulia rafikzeraoulia rafik
2,38711029
2,38711029
It should be $p+q=n-1$.
– user236182
Dec 12 '16 at 22:51
It's still impossible because $(p+q)-(p-q)=2q>0$ while $(n-1)-(n+1)=-2<0$.
– user236182
Dec 12 '16 at 22:54
for k=1 it's impossible and i cited your case p+q=n-1 in the side of n+q=p-1
– zeraoulia rafik
Dec 12 '16 at 22:55
If $p+1=n-q$, then $p+q=n-1$.
– user236182
Dec 12 '16 at 22:57
And if $n+q+1=p$, then $p-q=n+1$.
– user236182
Dec 12 '16 at 22:58
|
show 3 more comments
It should be $p+q=n-1$.
– user236182
Dec 12 '16 at 22:51
It's still impossible because $(p+q)-(p-q)=2q>0$ while $(n-1)-(n+1)=-2<0$.
– user236182
Dec 12 '16 at 22:54
for k=1 it's impossible and i cited your case p+q=n-1 in the side of n+q=p-1
– zeraoulia rafik
Dec 12 '16 at 22:55
If $p+1=n-q$, then $p+q=n-1$.
– user236182
Dec 12 '16 at 22:57
And if $n+q+1=p$, then $p-q=n+1$.
– user236182
Dec 12 '16 at 22:58
It should be $p+q=n-1$.
– user236182
Dec 12 '16 at 22:51
It should be $p+q=n-1$.
– user236182
Dec 12 '16 at 22:51
It's still impossible because $(p+q)-(p-q)=2q>0$ while $(n-1)-(n+1)=-2<0$.
– user236182
Dec 12 '16 at 22:54
It's still impossible because $(p+q)-(p-q)=2q>0$ while $(n-1)-(n+1)=-2<0$.
– user236182
Dec 12 '16 at 22:54
for k=1 it's impossible and i cited your case p+q=n-1 in the side of n+q=p-1
– zeraoulia rafik
Dec 12 '16 at 22:55
for k=1 it's impossible and i cited your case p+q=n-1 in the side of n+q=p-1
– zeraoulia rafik
Dec 12 '16 at 22:55
If $p+1=n-q$, then $p+q=n-1$.
– user236182
Dec 12 '16 at 22:57
If $p+1=n-q$, then $p+q=n-1$.
– user236182
Dec 12 '16 at 22:57
And if $n+q+1=p$, then $p-q=n+1$.
– user236182
Dec 12 '16 at 22:58
And if $n+q+1=p$, then $p-q=n+1$.
– user236182
Dec 12 '16 at 22:58
|
show 3 more comments
Case $pne q$: We can assume that $p>q$, then $$r^2+r leq (p-1)^2+p^2+p-1+p=2p^2implies boxed{rleq psqrt{2}}$$
From:
$$p(p+1) =r^2- q^2+r-q =(r-q)(r+q+1)$$
we get
If $pmid r-q$ then $r+q+1mid p+1$, so $pleq r-q<r$ and $r+q+1leq p+1 implies r<p$ a contradiction.
If $pmid r+q+1$ then $r-qmid p+1$. Since $r+q+1=kp$ for some integer $kgeq 1$ we have $$kpleq psqrt{2}+(p-1)+1 implies kleq 2$$
Case 1: $k=2$ we get $r+q+1=2p$ and $p+1=2r-2q$ from where we get $3p=11$, no good.
Case 2: $k=1$ we get $r+q+1=p$ and $p+1=r-q$ so $q=-1$ again contradiction.
So $p=q$ and now we have to solve $$2p^2+2p = r^2+r$$
So $$2p(p+1)=r(r+1)implies rmid 2p(p+1)$$
If $rmid 2$ then $r=2$ which is impossibile.
If $rmid p$ then $rleq p$ which is impossibile.
If $rmid p+1$ then $rleq p+1$ but then $r=p+1$ since $r>p$, so $r=3$ and $p=2$.
add a comment |
Case $pne q$: We can assume that $p>q$, then $$r^2+r leq (p-1)^2+p^2+p-1+p=2p^2implies boxed{rleq psqrt{2}}$$
From:
$$p(p+1) =r^2- q^2+r-q =(r-q)(r+q+1)$$
we get
If $pmid r-q$ then $r+q+1mid p+1$, so $pleq r-q<r$ and $r+q+1leq p+1 implies r<p$ a contradiction.
If $pmid r+q+1$ then $r-qmid p+1$. Since $r+q+1=kp$ for some integer $kgeq 1$ we have $$kpleq psqrt{2}+(p-1)+1 implies kleq 2$$
Case 1: $k=2$ we get $r+q+1=2p$ and $p+1=2r-2q$ from where we get $3p=11$, no good.
Case 2: $k=1$ we get $r+q+1=p$ and $p+1=r-q$ so $q=-1$ again contradiction.
So $p=q$ and now we have to solve $$2p^2+2p = r^2+r$$
So $$2p(p+1)=r(r+1)implies rmid 2p(p+1)$$
If $rmid 2$ then $r=2$ which is impossibile.
If $rmid p$ then $rleq p$ which is impossibile.
If $rmid p+1$ then $rleq p+1$ but then $r=p+1$ since $r>p$, so $r=3$ and $p=2$.
add a comment |
Case $pne q$: We can assume that $p>q$, then $$r^2+r leq (p-1)^2+p^2+p-1+p=2p^2implies boxed{rleq psqrt{2}}$$
From:
$$p(p+1) =r^2- q^2+r-q =(r-q)(r+q+1)$$
we get
If $pmid r-q$ then $r+q+1mid p+1$, so $pleq r-q<r$ and $r+q+1leq p+1 implies r<p$ a contradiction.
If $pmid r+q+1$ then $r-qmid p+1$. Since $r+q+1=kp$ for some integer $kgeq 1$ we have $$kpleq psqrt{2}+(p-1)+1 implies kleq 2$$
Case 1: $k=2$ we get $r+q+1=2p$ and $p+1=2r-2q$ from where we get $3p=11$, no good.
Case 2: $k=1$ we get $r+q+1=p$ and $p+1=r-q$ so $q=-1$ again contradiction.
So $p=q$ and now we have to solve $$2p^2+2p = r^2+r$$
So $$2p(p+1)=r(r+1)implies rmid 2p(p+1)$$
If $rmid 2$ then $r=2$ which is impossibile.
If $rmid p$ then $rleq p$ which is impossibile.
If $rmid p+1$ then $rleq p+1$ but then $r=p+1$ since $r>p$, so $r=3$ and $p=2$.
Case $pne q$: We can assume that $p>q$, then $$r^2+r leq (p-1)^2+p^2+p-1+p=2p^2implies boxed{rleq psqrt{2}}$$
From:
$$p(p+1) =r^2- q^2+r-q =(r-q)(r+q+1)$$
we get
If $pmid r-q$ then $r+q+1mid p+1$, so $pleq r-q<r$ and $r+q+1leq p+1 implies r<p$ a contradiction.
If $pmid r+q+1$ then $r-qmid p+1$. Since $r+q+1=kp$ for some integer $kgeq 1$ we have $$kpleq psqrt{2}+(p-1)+1 implies kleq 2$$
Case 1: $k=2$ we get $r+q+1=2p$ and $p+1=2r-2q$ from where we get $3p=11$, no good.
Case 2: $k=1$ we get $r+q+1=p$ and $p+1=r-q$ so $q=-1$ again contradiction.
So $p=q$ and now we have to solve $$2p^2+2p = r^2+r$$
So $$2p(p+1)=r(r+1)implies rmid 2p(p+1)$$
If $rmid 2$ then $r=2$ which is impossibile.
If $rmid p$ then $rleq p$ which is impossibile.
If $rmid p+1$ then $rleq p+1$ but then $r=p+1$ since $r>p$, so $r=3$ and $p=2$.
answered Nov 14 '18 at 23:13
greedoidgreedoid
38.5k114797
38.5k114797
add a comment |
add a comment |
From the given equation
$$p(p+1)+q(q+1)=r(r+1)$$
it follows that $p < r$ and $q < r$.
Next, another inequality which will be useful later . . .
Claim:$;p+q > r$.
Proof:
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(p^2+q^2)+(p+q)=r(r+1)\[4pt]
implies;&(p+q)^2+(p+q)>r(r+1)\[4pt]
implies;&(p+q)(p+q+1)>r(r+1)\[4pt]
implies;&p+q>r\[4pt]
end{align*}
as claimed.
Returning to the main problem . . .
First suppose $p=q$.
Then the given equation reduces to
$$2p(p+1)=r(r+1)$$
hence, since $r > p$, it follows that $r|(p+1)$.
But then $p < r le p+1$, so $r=p+1$, which implies $p=2$, and $r=3$.
It can be verified that the triple $(p,q,r)=(2,2,3)$ satisfies the given equation.
Next suppose $p,q$ are distinct.
Without loss of generality, assume $p < q$.
Suppose $;p=2$.
Then from $p < q < r$, we get $qge 3$ and $rge 5$, hence
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(2)(2+1)+q(q+1)=r(r+1)\[4pt]
implies;&r(r+1)-q(q+1)=6\[4pt]
implies;&(r-q)(q+r+1)=6\[4pt]
implies;&(q+r+1)mid 6\[4pt]
implies;&q+r+1le 6\[4pt]
end{align*}
contradiction, since $q+r+1ge 3+5+1=9$.
Hence we must have $p > 2$.
Since $p+q > r$, it follows that $pnotmid r-q$, and $qnotmid r-p$.
begin{align*}
text{Then};;&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&p(p+1)=r(r+1)-q(q+1)\[4pt]
implies;&p(p+1)=(q+r+1)(r-q)\[4pt]
implies;&pmid (q+r+1)\[4pt]
implies;&pmid (p+q+r+1)\[12pt]
text{and};,&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&q(q+1)=r(r+1)-p(p+1)\[4pt]
implies;&q(q+1)=(p+r+1)(r-p)\[4pt]
implies;&qmid (p+r+1)\[4pt]
implies;&qmid (p+q+r+1)\[12pt]
text{hence};,&pqmid (p+q+r+1)\[4pt]
implies;&pqle p+q+r+1\[4pt]
implies;&pq < p+q+(p+q)+1\[4pt]
implies;&pq-2p-2q < 1\[4pt]
implies;&(p-2)(q-2) < 5\[4pt]
implies;&q-2 < 5\[4pt]
implies;&q < 7\[4pt]
implies;&qle 5\[4pt]
implies;&(p,q)=(3,5)\[4pt]
implies;&(3)(3+1)+(5)(5+1)=r(r+1)\[4pt]
implies;&r=6\[4pt]
end{align*}
contradiction, since $6$ is not prime.
Therefore the only solution is $(p,q,r)=(2,2,3)$.
add a comment |
From the given equation
$$p(p+1)+q(q+1)=r(r+1)$$
it follows that $p < r$ and $q < r$.
Next, another inequality which will be useful later . . .
Claim:$;p+q > r$.
Proof:
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(p^2+q^2)+(p+q)=r(r+1)\[4pt]
implies;&(p+q)^2+(p+q)>r(r+1)\[4pt]
implies;&(p+q)(p+q+1)>r(r+1)\[4pt]
implies;&p+q>r\[4pt]
end{align*}
as claimed.
Returning to the main problem . . .
First suppose $p=q$.
Then the given equation reduces to
$$2p(p+1)=r(r+1)$$
hence, since $r > p$, it follows that $r|(p+1)$.
But then $p < r le p+1$, so $r=p+1$, which implies $p=2$, and $r=3$.
It can be verified that the triple $(p,q,r)=(2,2,3)$ satisfies the given equation.
Next suppose $p,q$ are distinct.
Without loss of generality, assume $p < q$.
Suppose $;p=2$.
Then from $p < q < r$, we get $qge 3$ and $rge 5$, hence
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(2)(2+1)+q(q+1)=r(r+1)\[4pt]
implies;&r(r+1)-q(q+1)=6\[4pt]
implies;&(r-q)(q+r+1)=6\[4pt]
implies;&(q+r+1)mid 6\[4pt]
implies;&q+r+1le 6\[4pt]
end{align*}
contradiction, since $q+r+1ge 3+5+1=9$.
Hence we must have $p > 2$.
Since $p+q > r$, it follows that $pnotmid r-q$, and $qnotmid r-p$.
begin{align*}
text{Then};;&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&p(p+1)=r(r+1)-q(q+1)\[4pt]
implies;&p(p+1)=(q+r+1)(r-q)\[4pt]
implies;&pmid (q+r+1)\[4pt]
implies;&pmid (p+q+r+1)\[12pt]
text{and};,&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&q(q+1)=r(r+1)-p(p+1)\[4pt]
implies;&q(q+1)=(p+r+1)(r-p)\[4pt]
implies;&qmid (p+r+1)\[4pt]
implies;&qmid (p+q+r+1)\[12pt]
text{hence};,&pqmid (p+q+r+1)\[4pt]
implies;&pqle p+q+r+1\[4pt]
implies;&pq < p+q+(p+q)+1\[4pt]
implies;&pq-2p-2q < 1\[4pt]
implies;&(p-2)(q-2) < 5\[4pt]
implies;&q-2 < 5\[4pt]
implies;&q < 7\[4pt]
implies;&qle 5\[4pt]
implies;&(p,q)=(3,5)\[4pt]
implies;&(3)(3+1)+(5)(5+1)=r(r+1)\[4pt]
implies;&r=6\[4pt]
end{align*}
contradiction, since $6$ is not prime.
Therefore the only solution is $(p,q,r)=(2,2,3)$.
add a comment |
From the given equation
$$p(p+1)+q(q+1)=r(r+1)$$
it follows that $p < r$ and $q < r$.
Next, another inequality which will be useful later . . .
Claim:$;p+q > r$.
Proof:
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(p^2+q^2)+(p+q)=r(r+1)\[4pt]
implies;&(p+q)^2+(p+q)>r(r+1)\[4pt]
implies;&(p+q)(p+q+1)>r(r+1)\[4pt]
implies;&p+q>r\[4pt]
end{align*}
as claimed.
Returning to the main problem . . .
First suppose $p=q$.
Then the given equation reduces to
$$2p(p+1)=r(r+1)$$
hence, since $r > p$, it follows that $r|(p+1)$.
But then $p < r le p+1$, so $r=p+1$, which implies $p=2$, and $r=3$.
It can be verified that the triple $(p,q,r)=(2,2,3)$ satisfies the given equation.
Next suppose $p,q$ are distinct.
Without loss of generality, assume $p < q$.
Suppose $;p=2$.
Then from $p < q < r$, we get $qge 3$ and $rge 5$, hence
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(2)(2+1)+q(q+1)=r(r+1)\[4pt]
implies;&r(r+1)-q(q+1)=6\[4pt]
implies;&(r-q)(q+r+1)=6\[4pt]
implies;&(q+r+1)mid 6\[4pt]
implies;&q+r+1le 6\[4pt]
end{align*}
contradiction, since $q+r+1ge 3+5+1=9$.
Hence we must have $p > 2$.
Since $p+q > r$, it follows that $pnotmid r-q$, and $qnotmid r-p$.
begin{align*}
text{Then};;&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&p(p+1)=r(r+1)-q(q+1)\[4pt]
implies;&p(p+1)=(q+r+1)(r-q)\[4pt]
implies;&pmid (q+r+1)\[4pt]
implies;&pmid (p+q+r+1)\[12pt]
text{and};,&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&q(q+1)=r(r+1)-p(p+1)\[4pt]
implies;&q(q+1)=(p+r+1)(r-p)\[4pt]
implies;&qmid (p+r+1)\[4pt]
implies;&qmid (p+q+r+1)\[12pt]
text{hence};,&pqmid (p+q+r+1)\[4pt]
implies;&pqle p+q+r+1\[4pt]
implies;&pq < p+q+(p+q)+1\[4pt]
implies;&pq-2p-2q < 1\[4pt]
implies;&(p-2)(q-2) < 5\[4pt]
implies;&q-2 < 5\[4pt]
implies;&q < 7\[4pt]
implies;&qle 5\[4pt]
implies;&(p,q)=(3,5)\[4pt]
implies;&(3)(3+1)+(5)(5+1)=r(r+1)\[4pt]
implies;&r=6\[4pt]
end{align*}
contradiction, since $6$ is not prime.
Therefore the only solution is $(p,q,r)=(2,2,3)$.
From the given equation
$$p(p+1)+q(q+1)=r(r+1)$$
it follows that $p < r$ and $q < r$.
Next, another inequality which will be useful later . . .
Claim:$;p+q > r$.
Proof:
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(p^2+q^2)+(p+q)=r(r+1)\[4pt]
implies;&(p+q)^2+(p+q)>r(r+1)\[4pt]
implies;&(p+q)(p+q+1)>r(r+1)\[4pt]
implies;&p+q>r\[4pt]
end{align*}
as claimed.
Returning to the main problem . . .
First suppose $p=q$.
Then the given equation reduces to
$$2p(p+1)=r(r+1)$$
hence, since $r > p$, it follows that $r|(p+1)$.
But then $p < r le p+1$, so $r=p+1$, which implies $p=2$, and $r=3$.
It can be verified that the triple $(p,q,r)=(2,2,3)$ satisfies the given equation.
Next suppose $p,q$ are distinct.
Without loss of generality, assume $p < q$.
Suppose $;p=2$.
Then from $p < q < r$, we get $qge 3$ and $rge 5$, hence
begin{align*}
&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&(2)(2+1)+q(q+1)=r(r+1)\[4pt]
implies;&r(r+1)-q(q+1)=6\[4pt]
implies;&(r-q)(q+r+1)=6\[4pt]
implies;&(q+r+1)mid 6\[4pt]
implies;&q+r+1le 6\[4pt]
end{align*}
contradiction, since $q+r+1ge 3+5+1=9$.
Hence we must have $p > 2$.
Since $p+q > r$, it follows that $pnotmid r-q$, and $qnotmid r-p$.
begin{align*}
text{Then};;&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&p(p+1)=r(r+1)-q(q+1)\[4pt]
implies;&p(p+1)=(q+r+1)(r-q)\[4pt]
implies;&pmid (q+r+1)\[4pt]
implies;&pmid (p+q+r+1)\[12pt]
text{and};,&p(p+1)+q(q+1)=r(r+1)\[4pt]
implies;&q(q+1)=r(r+1)-p(p+1)\[4pt]
implies;&q(q+1)=(p+r+1)(r-p)\[4pt]
implies;&qmid (p+r+1)\[4pt]
implies;&qmid (p+q+r+1)\[12pt]
text{hence};,&pqmid (p+q+r+1)\[4pt]
implies;&pqle p+q+r+1\[4pt]
implies;&pq < p+q+(p+q)+1\[4pt]
implies;&pq-2p-2q < 1\[4pt]
implies;&(p-2)(q-2) < 5\[4pt]
implies;&q-2 < 5\[4pt]
implies;&q < 7\[4pt]
implies;&qle 5\[4pt]
implies;&(p,q)=(3,5)\[4pt]
implies;&(3)(3+1)+(5)(5+1)=r(r+1)\[4pt]
implies;&r=6\[4pt]
end{align*}
contradiction, since $6$ is not prime.
Therefore the only solution is $(p,q,r)=(2,2,3)$.
edited Nov 14 '18 at 23:52
answered Nov 14 '18 at 23:37
quasiquasi
36k22662
36k22662
add a comment |
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f2056120%2fhow-do-i-find-all-prime-solutions-p-q-r-of-the-equation-displaystyle-pp1%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
How do you know that no other solutions exist then, what is your source ?
– Peter
Dec 12 '16 at 22:28
I think this problem related to A shinzel solution of the titled equation if i'm true
– zeraoulia rafik
Dec 12 '16 at 22:43
It may or may not help to rephrase it as $p^2 + p + q^2 + q = r^2 + r$.
– Robert Soupe
Dec 13 '16 at 2:21
this equivalent to solve $$(2p+1)^2+(2q+1)^2=(2r+1)^2+1$$
– Bumblebee
Jan 18 '18 at 20:01