Solutions to $a (2 a^2 + 2 b^2 + c^2 + d^2) = (2 a^3 + 2 b^3 + c^3 + d^3)$ in integers
-1
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I ask for positive integral solutions to $a (2 a^2 + 2 b^2 + c^2 + d^2) = (2 a^3 + 2 b^3 + c^3 + d^3)$, where a,b,c,d are positive integers and $aneq bneq cneq d$.
In particular does a solution exist for $d>1$.
number-theory
asked Jan 5 at 18:49
user631773user631773
11
$endgroup$
add a comment |
-1
$begingroup$
I ask for positive integral solutions to $a (2 a^2 + 2 b^2 + c^2 + d^2) = (2 a^3 + 2 b^3 + c^3 + d^3)$, where a,b,c,d are positive integers and $aneq bneq cneq d$.
In particular does a solution exist for $d>1$.
number-theory
asked Jan 5 at 18:49
user631773user631773
11
$endgroup$
$begingroup$
Would you append an update to this instead of duplication?
$endgroup$
– metamorphy
Jan 5 at 18:52
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@metamorphy it is not a duplication
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– user631773
Jan 5 at 18:53
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Very similar to ... math.stackexchange.com/questions/3062146/…
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– Donald Splutterwit
Jan 5 at 19:11
add a comment |
-1
-1
-1
1
$begingroup$
I ask for positive integral solutions to $a (2 a^2 + 2 b^2 + c^2 + d^2) = (2 a^3 + 2 b^3 + c^3 + d^3)$, where a,b,c,d are positive integers and $aneq bneq cneq d$.
In particular does a solution exist for $d>1$.
number-theory
asked Jan 5 at 18:49
user631773user631773
11
$endgroup$
I ask for positive integral solutions to $a (2 a^2 + 2 b^2 + c^2 + d^2) = (2 a^3 + 2 b^3 + c^3 + d^3)$, where a,b,c,d are positive integers and $aneq bneq cneq d$.
In particular does a solution exist for $d>1$.
number-theory
number-theory
asked Jan 5 at 18:49
user631773user631773
11
asked Jan 5 at 18:49
user631773user631773
11
asked Jan 5 at 18:49
user631773user631773
11
asked Jan 5 at 18:49
user631773user631773
11
asked Jan 5 at 18:49
user631773user631773
11
11
$begingroup$
Would you append an update to this instead of duplication?
$endgroup$
– metamorphy
Jan 5 at 18:52
$begingroup$
@metamorphy it is not a duplication
$endgroup$
– user631773
Jan 5 at 18:53
$begingroup$
Very similar to ... math.stackexchange.com/questions/3062146/…
$endgroup$
– Donald Splutterwit
Jan 5 at 19:11
add a comment |
$begingroup$
Would you append an update to this instead of duplication?
$endgroup$
– metamorphy
Jan 5 at 18:52
$begingroup$
@metamorphy it is not a duplication
$endgroup$
– user631773
Jan 5 at 18:53
$begingroup$
Very similar to ... math.stackexchange.com/questions/3062146/…
$endgroup$
– Donald Splutterwit
Jan 5 at 19:11
$begingroup$
Would you append an update to this instead of duplication?
$endgroup$
– metamorphy
Jan 5 at 18:52
$begingroup$
Would you append an update to this instead of duplication?
$endgroup$
– metamorphy
Jan 5 at 18:52
$begingroup$
@metamorphy it is not a duplication
$endgroup$
– user631773
Jan 5 at 18:53
$begingroup$
@metamorphy it is not a duplication
$endgroup$
– user631773
Jan 5 at 18:53
$begingroup$
Very similar to ... math.stackexchange.com/questions/3062146/…
$endgroup$
– Donald Splutterwit
Jan 5 at 19:11
$begingroup$
Very similar to ... math.stackexchange.com/questions/3062146/…
$endgroup$
– Donald Splutterwit
Jan 5 at 19:11
add a comment |
2 Answers
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Yes, $(5,2,6,2)$ by brute force.
Start by noting that $a = frac{2b^3+c^3+d^3}{2b^2+c^2+d^2}$. Since we want $d>1$, let $ d = 2$.
First try, let $b = 1$. The whole thing simplifies to $a = frac{c^3+10}{c^2+6} = c + frac{10-6c}{c^2+6}$. Now, $c = 1$ won't give integer. For $c > 2$, if $a$ is integer, then $|10-6c|geq c^2 + 6$, i.e. $6c-10geq c^2+6$, which has no real solutions. Thus, $a$ is never a positive integer.
Now try $b = 2$. Then, $a = frac{c^3+24}{c^2+12} = c + frac{24-12c}{c^2+12}$. As before, $c = 1$ is not a solution. $c = 2$ is a solution of type $a=b=c=d$. If $c>2$, for $a$ to be an integer, we need $|24-12c|geq c^2+12$. Now, $12c-24 geq c^2+12$ iff $(c-6)^2 leq 0$ iff $c = 6$. One checks that this $c$ gives $a= 5$.
answered Jan 5 at 19:25
EnnarEnnar
14.5k32443
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0
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take any triple $b,c,d,$ with $gcd(b,c,d)=1,$ there is a smallest $k$ such that $(a,kb,kc,kd)$ is a solution with $a$ integral, while $gcd(a,kb,kc,kd) = 1.$ Also
$$ k = frac{2b^2+c^2+d^2}{gcd left( 2b^2+c^2+d^2 ;, ;2b^3+c^3+d^3 right)} $$
original 1 1 1 mult 1 gives 1 1 1 1
original 1 2 1 mult 7 gives 11 7 14 7
original 1 2 2 mult 5 gives 9 5 10 10
original 1 3 1 mult 2 gives 5 2 6 2
original 1 3 2 mult 15 gives 37 15 45 30
original 1 3 3 mult 5 gives 14 5 15 15
original 1 4 1 mult 19 gives 67 19 76 19
original 1 4 2 mult 11 gives 37 11 44 22
original 1 4 3 mult 9 gives 31 9 36 27
original 1 4 4 mult 17 gives 65 17 68 68
original 1 5 1 mult 7 gives 32 7 35 7
original 1 5 2 mult 31 gives 135 31 155 62
original 1 5 3 mult 18 gives 77 18 90 54
original 1 5 4 mult 43 gives 191 43 215 172
original 1 5 5 mult 13 gives 63 13 65 65
original 1 6 1 mult 13 gives 73 13 78 13
original 1 6 2 mult 21 gives 113 21 126 42
original 1 6 3 mult 47 gives 245 47 282 141
original 1 6 4 mult 9 gives 47 9 54 36
original 1 6 5 mult 9 gives 49 9 54 45
original 1 6 6 mult 37 gives 217 37 222 222
original 2 1 1 mult 5 gives 9 10 5 5
original 2 2 1 mult 13 gives 25 26 26 13
original 2 3 1 mult 9 gives 22 18 27 9
original 2 3 2 mult 7 gives 17 14 21 14
original 2 3 3 mult 13 gives 35 26 39 39
original 2 4 1 mult 25 gives 81 50 100 25
original 2 4 3 mult 33 gives 107 66 132 99
original 2 5 1 mult 17 gives 71 34 85 17
original 2 5 2 mult 37 gives 149 74 185 74
original 2 5 3 mult 1 gives 4 2 5 3
original 2 5 4 mult 49 gives 205 98 245 196
original 2 5 5 mult 29 gives 133 58 145 145
original 2 6 1 mult 45 gives 233 90 270 45
original 2 6 3 mult 53 gives 259 106 318 159
original 2 6 5 mult 23 gives 119 46 138 115
original 3 1 1 mult 5 gives 14 15 5 5
original 3 2 1 mult 23 gives 63 69 46 23
original 3 2 2 mult 13 gives 35 39 26 26
original 3 3 1 mult 14 gives 41 42 42 14
original 3 3 2 mult 31 gives 89 93 93 62
original 3 4 1 mult 5 gives 17 15 20 5
original 3 4 2 mult 19 gives 63 57 76 38
original 3 4 3 mult 43 gives 145 129 172 129
original 3 4 4 mult 25 gives 91 75 100 100
original 3 5 1 mult 11 gives 45 33 55 11
original 3 5 2 mult 47 gives 187 141 235 94
original 3 5 3 mult 26 gives 103 78 130 78
original 3 5 4 mult 59 gives 243 177 295 236
original 3 5 5 mult 17 gives 76 51 85 85
original 3 6 1 mult 55 gives 271 165 330 55
original 3 6 2 mult 29 gives 139 87 174 58
original 3 6 4 mult 35 gives 167 105 210 140
original 3 6 5 mult 1 gives 5 3 6 5
original 4 1 1 mult 17 gives 65 68 17 17
original 4 2 1 mult 37 gives 137 148 74 37
original 4 3 1 mult 7 gives 26 28 21 7
original 4 3 2 mult 45 gives 163 180 135 90
original 4 3 3 mult 25 gives 91 100 75 75
original 4 4 1 mult 49 gives 193 196 196 49
original 4 4 3 mult 19 gives 73 76 76 57
original 4 5 1 mult 29 gives 127 116 145 29
original 4 5 2 mult 61 gives 261 244 305 122
original 4 5 3 mult 33 gives 140 132 165 99
original 4 5 4 mult 73 gives 317 292 365 292
original 4 5 5 mult 41 gives 189 164 205 205
original 4 6 1 mult 1 gives 5 4 6 1
original 4 6 3 mult 11 gives 53 44 66 33
original 4 6 5 mult 93 gives 469 372 558 465
original 5 1 1 mult 13 gives 63 65 13 13
original 5 2 1 mult 55 gives 259 275 110 55
original 5 2 2 mult 29 gives 133 145 58 58
original 5 3 1 mult 30 gives 139 150 90 30
original 5 3 2 mult 21 gives 95 105 63 42
original 5 3 3 mult 17 gives 76 85 51 51
original 5 4 1 mult 67 gives 315 335 268 67
original 5 4 2 mult 5 gives 23 25 20 10
original 5 4 3 mult 75 gives 341 375 300 225
original 5 4 4 mult 41 gives 189 205 164 164
original 5 5 1 mult 19 gives 94 95 95 19
original 5 5 2 mult 79 gives 383 395 395 158
original 5 5 3 mult 14 gives 67 70 70 42
original 5 5 4 mult 91 gives 439 455 455 364
original 5 6 1 mult 87 gives 467 435 522 87
original 5 6 2 mult 15 gives 79 75 90 30
original 5 6 3 mult 95 gives 493 475 570 285
original 5 6 4 mult 51 gives 265 255 306 204
original 5 6 5 mult 37 gives 197 185 222 185
original 5 6 6 mult 61 gives 341 305 366 366
original 6 1 1 mult 37 gives 217 222 37 37
original 6 2 1 mult 11 gives 63 66 22 11
original 6 3 1 mult 41 gives 230 246 123 41
original 6 3 2 mult 85 gives 467 510 255 170
original 6 4 1 mult 89 gives 497 534 356 89
original 6 4 3 mult 97 gives 523 582 388 291
original 6 5 1 mult 49 gives 279 294 245 49
original 6 5 2 mult 101 gives 565 606 505 202
original 6 5 3 mult 53 gives 292 318 265 159
original 6 5 4 mult 113 gives 621 678 565 452
original 6 5 5 mult 61 gives 341 366 305 305
original 6 6 1 mult 109 gives 649 654 654 109
original 6 6 5 mult 133 gives 773 798 798 665
answered Jan 5 at 19:40
Will JagyWill Jagy
103k5101200
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add a comment |
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2 Answers
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2 Answers
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$begingroup$
Yes, $(5,2,6,2)$ by brute force.
Start by noting that $a = frac{2b^3+c^3+d^3}{2b^2+c^2+d^2}$. Since we want $d>1$, let $ d = 2$.
First try, let $b = 1$. The whole thing simplifies to $a = frac{c^3+10}{c^2+6} = c + frac{10-6c}{c^2+6}$. Now, $c = 1$ won't give integer. For $c > 2$, if $a$ is integer, then $|10-6c|geq c^2 + 6$, i.e. $6c-10geq c^2+6$, which has no real solutions. Thus, $a$ is never a positive integer.
Now try $b = 2$. Then, $a = frac{c^3+24}{c^2+12} = c + frac{24-12c}{c^2+12}$. As before, $c = 1$ is not a solution. $c = 2$ is a solution of type $a=b=c=d$. If $c>2$, for $a$ to be an integer, we need $|24-12c|geq c^2+12$. Now, $12c-24 geq c^2+12$ iff $(c-6)^2 leq 0$ iff $c = 6$. One checks that this $c$ gives $a= 5$.
answered Jan 5 at 19:25
EnnarEnnar
14.5k32443
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add a comment |
0
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Yes, $(5,2,6,2)$ by brute force.
Start by noting that $a = frac{2b^3+c^3+d^3}{2b^2+c^2+d^2}$. Since we want $d>1$, let $ d = 2$.
First try, let $b = 1$. The whole thing simplifies to $a = frac{c^3+10}{c^2+6} = c + frac{10-6c}{c^2+6}$. Now, $c = 1$ won't give integer. For $c > 2$, if $a$ is integer, then $|10-6c|geq c^2 + 6$, i.e. $6c-10geq c^2+6$, which has no real solutions. Thus, $a$ is never a positive integer.
Now try $b = 2$. Then, $a = frac{c^3+24}{c^2+12} = c + frac{24-12c}{c^2+12}$. As before, $c = 1$ is not a solution. $c = 2$ is a solution of type $a=b=c=d$. If $c>2$, for $a$ to be an integer, we need $|24-12c|geq c^2+12$. Now, $12c-24 geq c^2+12$ iff $(c-6)^2 leq 0$ iff $c = 6$. One checks that this $c$ gives $a= 5$.
answered Jan 5 at 19:25
EnnarEnnar
14.5k32443
$endgroup$
add a comment |
0
0
0
$begingroup$
Yes, $(5,2,6,2)$ by brute force.
Start by noting that $a = frac{2b^3+c^3+d^3}{2b^2+c^2+d^2}$. Since we want $d>1$, let $ d = 2$.
First try, let $b = 1$. The whole thing simplifies to $a = frac{c^3+10}{c^2+6} = c + frac{10-6c}{c^2+6}$. Now, $c = 1$ won't give integer. For $c > 2$, if $a$ is integer, then $|10-6c|geq c^2 + 6$, i.e. $6c-10geq c^2+6$, which has no real solutions. Thus, $a$ is never a positive integer.
Now try $b = 2$. Then, $a = frac{c^3+24}{c^2+12} = c + frac{24-12c}{c^2+12}$. As before, $c = 1$ is not a solution. $c = 2$ is a solution of type $a=b=c=d$. If $c>2$, for $a$ to be an integer, we need $|24-12c|geq c^2+12$. Now, $12c-24 geq c^2+12$ iff $(c-6)^2 leq 0$ iff $c = 6$. One checks that this $c$ gives $a= 5$.
answered Jan 5 at 19:25
EnnarEnnar
14.5k32443
$endgroup$
Yes, $(5,2,6,2)$ by brute force.
Start by noting that $a = frac{2b^3+c^3+d^3}{2b^2+c^2+d^2}$. Since we want $d>1$, let $ d = 2$.
First try, let $b = 1$. The whole thing simplifies to $a = frac{c^3+10}{c^2+6} = c + frac{10-6c}{c^2+6}$. Now, $c = 1$ won't give integer. For $c > 2$, if $a$ is integer, then $|10-6c|geq c^2 + 6$, i.e. $6c-10geq c^2+6$, which has no real solutions. Thus, $a$ is never a positive integer.
Now try $b = 2$. Then, $a = frac{c^3+24}{c^2+12} = c + frac{24-12c}{c^2+12}$. As before, $c = 1$ is not a solution. $c = 2$ is a solution of type $a=b=c=d$. If $c>2$, for $a$ to be an integer, we need $|24-12c|geq c^2+12$. Now, $12c-24 geq c^2+12$ iff $(c-6)^2 leq 0$ iff $c = 6$. One checks that this $c$ gives $a= 5$.
answered Jan 5 at 19:25
EnnarEnnar
14.5k32443
answered Jan 5 at 19:25
EnnarEnnar
14.5k32443
answered Jan 5 at 19:25
EnnarEnnar
14.5k32443
answered Jan 5 at 19:25
EnnarEnnar
14.5k32443
14.5k32443
add a comment |
add a comment |
0
$begingroup$
take any triple $b,c,d,$ with $gcd(b,c,d)=1,$ there is a smallest $k$ such that $(a,kb,kc,kd)$ is a solution with $a$ integral, while $gcd(a,kb,kc,kd) = 1.$ Also
$$ k = frac{2b^2+c^2+d^2}{gcd left( 2b^2+c^2+d^2 ;, ;2b^3+c^3+d^3 right)} $$
original 1 1 1 mult 1 gives 1 1 1 1
original 1 2 1 mult 7 gives 11 7 14 7
original 1 2 2 mult 5 gives 9 5 10 10
original 1 3 1 mult 2 gives 5 2 6 2
original 1 3 2 mult 15 gives 37 15 45 30
original 1 3 3 mult 5 gives 14 5 15 15
original 1 4 1 mult 19 gives 67 19 76 19
original 1 4 2 mult 11 gives 37 11 44 22
original 1 4 3 mult 9 gives 31 9 36 27
original 1 4 4 mult 17 gives 65 17 68 68
original 1 5 1 mult 7 gives 32 7 35 7
original 1 5 2 mult 31 gives 135 31 155 62
original 1 5 3 mult 18 gives 77 18 90 54
original 1 5 4 mult 43 gives 191 43 215 172
original 1 5 5 mult 13 gives 63 13 65 65
original 1 6 1 mult 13 gives 73 13 78 13
original 1 6 2 mult 21 gives 113 21 126 42
original 1 6 3 mult 47 gives 245 47 282 141
original 1 6 4 mult 9 gives 47 9 54 36
original 1 6 5 mult 9 gives 49 9 54 45
original 1 6 6 mult 37 gives 217 37 222 222
original 2 1 1 mult 5 gives 9 10 5 5
original 2 2 1 mult 13 gives 25 26 26 13
original 2 3 1 mult 9 gives 22 18 27 9
original 2 3 2 mult 7 gives 17 14 21 14
original 2 3 3 mult 13 gives 35 26 39 39
original 2 4 1 mult 25 gives 81 50 100 25
original 2 4 3 mult 33 gives 107 66 132 99
original 2 5 1 mult 17 gives 71 34 85 17
original 2 5 2 mult 37 gives 149 74 185 74
original 2 5 3 mult 1 gives 4 2 5 3
original 2 5 4 mult 49 gives 205 98 245 196
original 2 5 5 mult 29 gives 133 58 145 145
original 2 6 1 mult 45 gives 233 90 270 45
original 2 6 3 mult 53 gives 259 106 318 159
original 2 6 5 mult 23 gives 119 46 138 115
original 3 1 1 mult 5 gives 14 15 5 5
original 3 2 1 mult 23 gives 63 69 46 23
original 3 2 2 mult 13 gives 35 39 26 26
original 3 3 1 mult 14 gives 41 42 42 14
original 3 3 2 mult 31 gives 89 93 93 62
original 3 4 1 mult 5 gives 17 15 20 5
original 3 4 2 mult 19 gives 63 57 76 38
original 3 4 3 mult 43 gives 145 129 172 129
original 3 4 4 mult 25 gives 91 75 100 100
original 3 5 1 mult 11 gives 45 33 55 11
original 3 5 2 mult 47 gives 187 141 235 94
original 3 5 3 mult 26 gives 103 78 130 78
original 3 5 4 mult 59 gives 243 177 295 236
original 3 5 5 mult 17 gives 76 51 85 85
original 3 6 1 mult 55 gives 271 165 330 55
original 3 6 2 mult 29 gives 139 87 174 58
original 3 6 4 mult 35 gives 167 105 210 140
original 3 6 5 mult 1 gives 5 3 6 5
original 4 1 1 mult 17 gives 65 68 17 17
original 4 2 1 mult 37 gives 137 148 74 37
original 4 3 1 mult 7 gives 26 28 21 7
original 4 3 2 mult 45 gives 163 180 135 90
original 4 3 3 mult 25 gives 91 100 75 75
original 4 4 1 mult 49 gives 193 196 196 49
original 4 4 3 mult 19 gives 73 76 76 57
original 4 5 1 mult 29 gives 127 116 145 29
original 4 5 2 mult 61 gives 261 244 305 122
original 4 5 3 mult 33 gives 140 132 165 99
original 4 5 4 mult 73 gives 317 292 365 292
original 4 5 5 mult 41 gives 189 164 205 205
original 4 6 1 mult 1 gives 5 4 6 1
original 4 6 3 mult 11 gives 53 44 66 33
original 4 6 5 mult 93 gives 469 372 558 465
original 5 1 1 mult 13 gives 63 65 13 13
original 5 2 1 mult 55 gives 259 275 110 55
original 5 2 2 mult 29 gives 133 145 58 58
original 5 3 1 mult 30 gives 139 150 90 30
original 5 3 2 mult 21 gives 95 105 63 42
original 5 3 3 mult 17 gives 76 85 51 51
original 5 4 1 mult 67 gives 315 335 268 67
original 5 4 2 mult 5 gives 23 25 20 10
original 5 4 3 mult 75 gives 341 375 300 225
original 5 4 4 mult 41 gives 189 205 164 164
original 5 5 1 mult 19 gives 94 95 95 19
original 5 5 2 mult 79 gives 383 395 395 158
original 5 5 3 mult 14 gives 67 70 70 42
original 5 5 4 mult 91 gives 439 455 455 364
original 5 6 1 mult 87 gives 467 435 522 87
original 5 6 2 mult 15 gives 79 75 90 30
original 5 6 3 mult 95 gives 493 475 570 285
original 5 6 4 mult 51 gives 265 255 306 204
original 5 6 5 mult 37 gives 197 185 222 185
original 5 6 6 mult 61 gives 341 305 366 366
original 6 1 1 mult 37 gives 217 222 37 37
original 6 2 1 mult 11 gives 63 66 22 11
original 6 3 1 mult 41 gives 230 246 123 41
original 6 3 2 mult 85 gives 467 510 255 170
original 6 4 1 mult 89 gives 497 534 356 89
original 6 4 3 mult 97 gives 523 582 388 291
original 6 5 1 mult 49 gives 279 294 245 49
original 6 5 2 mult 101 gives 565 606 505 202
original 6 5 3 mult 53 gives 292 318 265 159
original 6 5 4 mult 113 gives 621 678 565 452
original 6 5 5 mult 61 gives 341 366 305 305
original 6 6 1 mult 109 gives 649 654 654 109
original 6 6 5 mult 133 gives 773 798 798 665
answered Jan 5 at 19:40
Will JagyWill Jagy
103k5101200
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add a comment |
0
$begingroup$
take any triple $b,c,d,$ with $gcd(b,c,d)=1,$ there is a smallest $k$ such that $(a,kb,kc,kd)$ is a solution with $a$ integral, while $gcd(a,kb,kc,kd) = 1.$ Also
$$ k = frac{2b^2+c^2+d^2}{gcd left( 2b^2+c^2+d^2 ;, ;2b^3+c^3+d^3 right)} $$
original 1 1 1 mult 1 gives 1 1 1 1
original 1 2 1 mult 7 gives 11 7 14 7
original 1 2 2 mult 5 gives 9 5 10 10
original 1 3 1 mult 2 gives 5 2 6 2
original 1 3 2 mult 15 gives 37 15 45 30
original 1 3 3 mult 5 gives 14 5 15 15
original 1 4 1 mult 19 gives 67 19 76 19
original 1 4 2 mult 11 gives 37 11 44 22
original 1 4 3 mult 9 gives 31 9 36 27
original 1 4 4 mult 17 gives 65 17 68 68
original 1 5 1 mult 7 gives 32 7 35 7
original 1 5 2 mult 31 gives 135 31 155 62
original 1 5 3 mult 18 gives 77 18 90 54
original 1 5 4 mult 43 gives 191 43 215 172
original 1 5 5 mult 13 gives 63 13 65 65
original 1 6 1 mult 13 gives 73 13 78 13
original 1 6 2 mult 21 gives 113 21 126 42
original 1 6 3 mult 47 gives 245 47 282 141
original 1 6 4 mult 9 gives 47 9 54 36
original 1 6 5 mult 9 gives 49 9 54 45
original 1 6 6 mult 37 gives 217 37 222 222
original 2 1 1 mult 5 gives 9 10 5 5
original 2 2 1 mult 13 gives 25 26 26 13
original 2 3 1 mult 9 gives 22 18 27 9
original 2 3 2 mult 7 gives 17 14 21 14
original 2 3 3 mult 13 gives 35 26 39 39
original 2 4 1 mult 25 gives 81 50 100 25
original 2 4 3 mult 33 gives 107 66 132 99
original 2 5 1 mult 17 gives 71 34 85 17
original 2 5 2 mult 37 gives 149 74 185 74
original 2 5 3 mult 1 gives 4 2 5 3
original 2 5 4 mult 49 gives 205 98 245 196
original 2 5 5 mult 29 gives 133 58 145 145
original 2 6 1 mult 45 gives 233 90 270 45
original 2 6 3 mult 53 gives 259 106 318 159
original 2 6 5 mult 23 gives 119 46 138 115
original 3 1 1 mult 5 gives 14 15 5 5
original 3 2 1 mult 23 gives 63 69 46 23
original 3 2 2 mult 13 gives 35 39 26 26
original 3 3 1 mult 14 gives 41 42 42 14
original 3 3 2 mult 31 gives 89 93 93 62
original 3 4 1 mult 5 gives 17 15 20 5
original 3 4 2 mult 19 gives 63 57 76 38
original 3 4 3 mult 43 gives 145 129 172 129
original 3 4 4 mult 25 gives 91 75 100 100
original 3 5 1 mult 11 gives 45 33 55 11
original 3 5 2 mult 47 gives 187 141 235 94
original 3 5 3 mult 26 gives 103 78 130 78
original 3 5 4 mult 59 gives 243 177 295 236
original 3 5 5 mult 17 gives 76 51 85 85
original 3 6 1 mult 55 gives 271 165 330 55
original 3 6 2 mult 29 gives 139 87 174 58
original 3 6 4 mult 35 gives 167 105 210 140
original 3 6 5 mult 1 gives 5 3 6 5
original 4 1 1 mult 17 gives 65 68 17 17
original 4 2 1 mult 37 gives 137 148 74 37
original 4 3 1 mult 7 gives 26 28 21 7
original 4 3 2 mult 45 gives 163 180 135 90
original 4 3 3 mult 25 gives 91 100 75 75
original 4 4 1 mult 49 gives 193 196 196 49
original 4 4 3 mult 19 gives 73 76 76 57
original 4 5 1 mult 29 gives 127 116 145 29
original 4 5 2 mult 61 gives 261 244 305 122
original 4 5 3 mult 33 gives 140 132 165 99
original 4 5 4 mult 73 gives 317 292 365 292
original 4 5 5 mult 41 gives 189 164 205 205
original 4 6 1 mult 1 gives 5 4 6 1
original 4 6 3 mult 11 gives 53 44 66 33
original 4 6 5 mult 93 gives 469 372 558 465
original 5 1 1 mult 13 gives 63 65 13 13
original 5 2 1 mult 55 gives 259 275 110 55
original 5 2 2 mult 29 gives 133 145 58 58
original 5 3 1 mult 30 gives 139 150 90 30
original 5 3 2 mult 21 gives 95 105 63 42
original 5 3 3 mult 17 gives 76 85 51 51
original 5 4 1 mult 67 gives 315 335 268 67
original 5 4 2 mult 5 gives 23 25 20 10
original 5 4 3 mult 75 gives 341 375 300 225
original 5 4 4 mult 41 gives 189 205 164 164
original 5 5 1 mult 19 gives 94 95 95 19
original 5 5 2 mult 79 gives 383 395 395 158
original 5 5 3 mult 14 gives 67 70 70 42
original 5 5 4 mult 91 gives 439 455 455 364
original 5 6 1 mult 87 gives 467 435 522 87
original 5 6 2 mult 15 gives 79 75 90 30
original 5 6 3 mult 95 gives 493 475 570 285
original 5 6 4 mult 51 gives 265 255 306 204
original 5 6 5 mult 37 gives 197 185 222 185
original 5 6 6 mult 61 gives 341 305 366 366
original 6 1 1 mult 37 gives 217 222 37 37
original 6 2 1 mult 11 gives 63 66 22 11
original 6 3 1 mult 41 gives 230 246 123 41
original 6 3 2 mult 85 gives 467 510 255 170
original 6 4 1 mult 89 gives 497 534 356 89
original 6 4 3 mult 97 gives 523 582 388 291
original 6 5 1 mult 49 gives 279 294 245 49
original 6 5 2 mult 101 gives 565 606 505 202
original 6 5 3 mult 53 gives 292 318 265 159
original 6 5 4 mult 113 gives 621 678 565 452
original 6 5 5 mult 61 gives 341 366 305 305
original 6 6 1 mult 109 gives 649 654 654 109
original 6 6 5 mult 133 gives 773 798 798 665
answered Jan 5 at 19:40
Will JagyWill Jagy
103k5101200
$endgroup$
add a comment |
0
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$begingroup$
take any triple $b,c,d,$ with $gcd(b,c,d)=1,$ there is a smallest $k$ such that $(a,kb,kc,kd)$ is a solution with $a$ integral, while $gcd(a,kb,kc,kd) = 1.$ Also
$$ k = frac{2b^2+c^2+d^2}{gcd left( 2b^2+c^2+d^2 ;, ;2b^3+c^3+d^3 right)} $$
original 1 1 1 mult 1 gives 1 1 1 1
original 1 2 1 mult 7 gives 11 7 14 7
original 1 2 2 mult 5 gives 9 5 10 10
original 1 3 1 mult 2 gives 5 2 6 2
original 1 3 2 mult 15 gives 37 15 45 30
original 1 3 3 mult 5 gives 14 5 15 15
original 1 4 1 mult 19 gives 67 19 76 19
original 1 4 2 mult 11 gives 37 11 44 22
original 1 4 3 mult 9 gives 31 9 36 27
original 1 4 4 mult 17 gives 65 17 68 68
original 1 5 1 mult 7 gives 32 7 35 7
original 1 5 2 mult 31 gives 135 31 155 62
original 1 5 3 mult 18 gives 77 18 90 54
original 1 5 4 mult 43 gives 191 43 215 172
original 1 5 5 mult 13 gives 63 13 65 65
original 1 6 1 mult 13 gives 73 13 78 13
original 1 6 2 mult 21 gives 113 21 126 42
original 1 6 3 mult 47 gives 245 47 282 141
original 1 6 4 mult 9 gives 47 9 54 36
original 1 6 5 mult 9 gives 49 9 54 45
original 1 6 6 mult 37 gives 217 37 222 222
original 2 1 1 mult 5 gives 9 10 5 5
original 2 2 1 mult 13 gives 25 26 26 13
original 2 3 1 mult 9 gives 22 18 27 9
original 2 3 2 mult 7 gives 17 14 21 14
original 2 3 3 mult 13 gives 35 26 39 39
original 2 4 1 mult 25 gives 81 50 100 25
original 2 4 3 mult 33 gives 107 66 132 99
original 2 5 1 mult 17 gives 71 34 85 17
original 2 5 2 mult 37 gives 149 74 185 74
original 2 5 3 mult 1 gives 4 2 5 3
original 2 5 4 mult 49 gives 205 98 245 196
original 2 5 5 mult 29 gives 133 58 145 145
original 2 6 1 mult 45 gives 233 90 270 45
original 2 6 3 mult 53 gives 259 106 318 159
original 2 6 5 mult 23 gives 119 46 138 115
original 3 1 1 mult 5 gives 14 15 5 5
original 3 2 1 mult 23 gives 63 69 46 23
original 3 2 2 mult 13 gives 35 39 26 26
original 3 3 1 mult 14 gives 41 42 42 14
original 3 3 2 mult 31 gives 89 93 93 62
original 3 4 1 mult 5 gives 17 15 20 5
original 3 4 2 mult 19 gives 63 57 76 38
original 3 4 3 mult 43 gives 145 129 172 129
original 3 4 4 mult 25 gives 91 75 100 100
original 3 5 1 mult 11 gives 45 33 55 11
original 3 5 2 mult 47 gives 187 141 235 94
original 3 5 3 mult 26 gives 103 78 130 78
original 3 5 4 mult 59 gives 243 177 295 236
original 3 5 5 mult 17 gives 76 51 85 85
original 3 6 1 mult 55 gives 271 165 330 55
original 3 6 2 mult 29 gives 139 87 174 58
original 3 6 4 mult 35 gives 167 105 210 140
original 3 6 5 mult 1 gives 5 3 6 5
original 4 1 1 mult 17 gives 65 68 17 17
original 4 2 1 mult 37 gives 137 148 74 37
original 4 3 1 mult 7 gives 26 28 21 7
original 4 3 2 mult 45 gives 163 180 135 90
original 4 3 3 mult 25 gives 91 100 75 75
original 4 4 1 mult 49 gives 193 196 196 49
original 4 4 3 mult 19 gives 73 76 76 57
original 4 5 1 mult 29 gives 127 116 145 29
original 4 5 2 mult 61 gives 261 244 305 122
original 4 5 3 mult 33 gives 140 132 165 99
original 4 5 4 mult 73 gives 317 292 365 292
original 4 5 5 mult 41 gives 189 164 205 205
original 4 6 1 mult 1 gives 5 4 6 1
original 4 6 3 mult 11 gives 53 44 66 33
original 4 6 5 mult 93 gives 469 372 558 465
original 5 1 1 mult 13 gives 63 65 13 13
original 5 2 1 mult 55 gives 259 275 110 55
original 5 2 2 mult 29 gives 133 145 58 58
original 5 3 1 mult 30 gives 139 150 90 30
original 5 3 2 mult 21 gives 95 105 63 42
original 5 3 3 mult 17 gives 76 85 51 51
original 5 4 1 mult 67 gives 315 335 268 67
original 5 4 2 mult 5 gives 23 25 20 10
original 5 4 3 mult 75 gives 341 375 300 225
original 5 4 4 mult 41 gives 189 205 164 164
original 5 5 1 mult 19 gives 94 95 95 19
original 5 5 2 mult 79 gives 383 395 395 158
original 5 5 3 mult 14 gives 67 70 70 42
original 5 5 4 mult 91 gives 439 455 455 364
original 5 6 1 mult 87 gives 467 435 522 87
original 5 6 2 mult 15 gives 79 75 90 30
original 5 6 3 mult 95 gives 493 475 570 285
original 5 6 4 mult 51 gives 265 255 306 204
original 5 6 5 mult 37 gives 197 185 222 185
original 5 6 6 mult 61 gives 341 305 366 366
original 6 1 1 mult 37 gives 217 222 37 37
original 6 2 1 mult 11 gives 63 66 22 11
original 6 3 1 mult 41 gives 230 246 123 41
original 6 3 2 mult 85 gives 467 510 255 170
original 6 4 1 mult 89 gives 497 534 356 89
original 6 4 3 mult 97 gives 523 582 388 291
original 6 5 1 mult 49 gives 279 294 245 49
original 6 5 2 mult 101 gives 565 606 505 202
original 6 5 3 mult 53 gives 292 318 265 159
original 6 5 4 mult 113 gives 621 678 565 452
original 6 5 5 mult 61 gives 341 366 305 305
original 6 6 1 mult 109 gives 649 654 654 109
original 6 6 5 mult 133 gives 773 798 798 665
answered Jan 5 at 19:40
Will JagyWill Jagy
103k5101200
$endgroup$
take any triple $b,c,d,$ with $gcd(b,c,d)=1,$ there is a smallest $k$ such that $(a,kb,kc,kd)$ is a solution with $a$ integral, while $gcd(a,kb,kc,kd) = 1.$ Also
$$ k = frac{2b^2+c^2+d^2}{gcd left( 2b^2+c^2+d^2 ;, ;2b^3+c^3+d^3 right)} $$
original 1 1 1 mult 1 gives 1 1 1 1
original 1 2 1 mult 7 gives 11 7 14 7
original 1 2 2 mult 5 gives 9 5 10 10
original 1 3 1 mult 2 gives 5 2 6 2
original 1 3 2 mult 15 gives 37 15 45 30
original 1 3 3 mult 5 gives 14 5 15 15
original 1 4 1 mult 19 gives 67 19 76 19
original 1 4 2 mult 11 gives 37 11 44 22
original 1 4 3 mult 9 gives 31 9 36 27
original 1 4 4 mult 17 gives 65 17 68 68
original 1 5 1 mult 7 gives 32 7 35 7
original 1 5 2 mult 31 gives 135 31 155 62
original 1 5 3 mult 18 gives 77 18 90 54
original 1 5 4 mult 43 gives 191 43 215 172
original 1 5 5 mult 13 gives 63 13 65 65
original 1 6 1 mult 13 gives 73 13 78 13
original 1 6 2 mult 21 gives 113 21 126 42
original 1 6 3 mult 47 gives 245 47 282 141
original 1 6 4 mult 9 gives 47 9 54 36
original 1 6 5 mult 9 gives 49 9 54 45
original 1 6 6 mult 37 gives 217 37 222 222
original 2 1 1 mult 5 gives 9 10 5 5
original 2 2 1 mult 13 gives 25 26 26 13
original 2 3 1 mult 9 gives 22 18 27 9
original 2 3 2 mult 7 gives 17 14 21 14
original 2 3 3 mult 13 gives 35 26 39 39
original 2 4 1 mult 25 gives 81 50 100 25
original 2 4 3 mult 33 gives 107 66 132 99
original 2 5 1 mult 17 gives 71 34 85 17
original 2 5 2 mult 37 gives 149 74 185 74
original 2 5 3 mult 1 gives 4 2 5 3
original 2 5 4 mult 49 gives 205 98 245 196
original 2 5 5 mult 29 gives 133 58 145 145
original 2 6 1 mult 45 gives 233 90 270 45
original 2 6 3 mult 53 gives 259 106 318 159
original 2 6 5 mult 23 gives 119 46 138 115
original 3 1 1 mult 5 gives 14 15 5 5
original 3 2 1 mult 23 gives 63 69 46 23
original 3 2 2 mult 13 gives 35 39 26 26
original 3 3 1 mult 14 gives 41 42 42 14
original 3 3 2 mult 31 gives 89 93 93 62
original 3 4 1 mult 5 gives 17 15 20 5
original 3 4 2 mult 19 gives 63 57 76 38
original 3 4 3 mult 43 gives 145 129 172 129
original 3 4 4 mult 25 gives 91 75 100 100
original 3 5 1 mult 11 gives 45 33 55 11
original 3 5 2 mult 47 gives 187 141 235 94
original 3 5 3 mult 26 gives 103 78 130 78
original 3 5 4 mult 59 gives 243 177 295 236
original 3 5 5 mult 17 gives 76 51 85 85
original 3 6 1 mult 55 gives 271 165 330 55
original 3 6 2 mult 29 gives 139 87 174 58
original 3 6 4 mult 35 gives 167 105 210 140
original 3 6 5 mult 1 gives 5 3 6 5
original 4 1 1 mult 17 gives 65 68 17 17
original 4 2 1 mult 37 gives 137 148 74 37
original 4 3 1 mult 7 gives 26 28 21 7
original 4 3 2 mult 45 gives 163 180 135 90
original 4 3 3 mult 25 gives 91 100 75 75
original 4 4 1 mult 49 gives 193 196 196 49
original 4 4 3 mult 19 gives 73 76 76 57
original 4 5 1 mult 29 gives 127 116 145 29
original 4 5 2 mult 61 gives 261 244 305 122
original 4 5 3 mult 33 gives 140 132 165 99
original 4 5 4 mult 73 gives 317 292 365 292
original 4 5 5 mult 41 gives 189 164 205 205
original 4 6 1 mult 1 gives 5 4 6 1
original 4 6 3 mult 11 gives 53 44 66 33
original 4 6 5 mult 93 gives 469 372 558 465
original 5 1 1 mult 13 gives 63 65 13 13
original 5 2 1 mult 55 gives 259 275 110 55
original 5 2 2 mult 29 gives 133 145 58 58
original 5 3 1 mult 30 gives 139 150 90 30
original 5 3 2 mult 21 gives 95 105 63 42
original 5 3 3 mult 17 gives 76 85 51 51
original 5 4 1 mult 67 gives 315 335 268 67
original 5 4 2 mult 5 gives 23 25 20 10
original 5 4 3 mult 75 gives 341 375 300 225
original 5 4 4 mult 41 gives 189 205 164 164
original 5 5 1 mult 19 gives 94 95 95 19
original 5 5 2 mult 79 gives 383 395 395 158
original 5 5 3 mult 14 gives 67 70 70 42
original 5 5 4 mult 91 gives 439 455 455 364
original 5 6 1 mult 87 gives 467 435 522 87
original 5 6 2 mult 15 gives 79 75 90 30
original 5 6 3 mult 95 gives 493 475 570 285
original 5 6 4 mult 51 gives 265 255 306 204
original 5 6 5 mult 37 gives 197 185 222 185
original 5 6 6 mult 61 gives 341 305 366 366
original 6 1 1 mult 37 gives 217 222 37 37
original 6 2 1 mult 11 gives 63 66 22 11
original 6 3 1 mult 41 gives 230 246 123 41
original 6 3 2 mult 85 gives 467 510 255 170
original 6 4 1 mult 89 gives 497 534 356 89
original 6 4 3 mult 97 gives 523 582 388 291
original 6 5 1 mult 49 gives 279 294 245 49
original 6 5 2 mult 101 gives 565 606 505 202
original 6 5 3 mult 53 gives 292 318 265 159
original 6 5 4 mult 113 gives 621 678 565 452
original 6 5 5 mult 61 gives 341 366 305 305
original 6 6 1 mult 109 gives 649 654 654 109
original 6 6 5 mult 133 gives 773 798 798 665
answered Jan 5 at 19:40
Will JagyWill Jagy
103k5101200
edited Jan 5 at 20:01
answered Jan 5 at 19:40
Will JagyWill Jagy
103k5101200
answered Jan 5 at 19:40
Will JagyWill Jagy
103k5101200
answered Jan 5 at 19:40
Will JagyWill Jagy
103k5101200
103k5101200
add a comment |
add a comment |
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$begingroup$
Would you append an update to this instead of duplication?
$endgroup$
– metamorphy
Jan 5 at 18:52
$begingroup$
@metamorphy it is not a duplication
$endgroup$
– user631773
Jan 5 at 18:53
$begingroup$
Very similar to ... math.stackexchange.com/questions/3062146/…
$endgroup$
– Donald Splutterwit
Jan 5 at 19:11