- Thread starter
- #1

#### mente oscura

##### Well-known member

- Nov 29, 2013

- 172

A simple question.

Solve in all positive integers, for:

[tex]a^2=9555^2+c^2[/tex]

Please, you show the way of solving it.

Regards.

- Thread starter mente oscura
- Start date

- Thread starter
- #1

- Nov 29, 2013

- 172

A simple question.

Solve in all positive integers, for:

[tex]a^2=9555^2+c^2[/tex]

Please, you show the way of solving it.

Regards.

- Aug 18, 2013

- 76

$$a^2-c^2=9555^2$$

$$(a-c)(a+c)=9555^2$$

If we let $f$ be a factor of $9555^2$, then we can let $a-c=f$ and $a+c=\frac{9555^2}{f}$, and $a=\frac{1}{2}(f+\frac{9555^2}{f})$. Now, since all the factors of $9555$ are odd, then $a$ will always be an integer. $9555$ has $24$ factors, hence there are $12$ possible pairs of $(a,c)$.

- Admin
- #3

- Feb 14, 2012

- 3,815

Hey

First notice that $a^2=9555^2+c^2$ can be rewritten in the following form:

$a^2-c^2=9555^2$

$(a+c)(a-c)=9555^2$

Since $9555=3\cdot5\cdot7\cdot7\cdot13$, if we are to arrange this number into the product of two square numbers, we see that we can have a total of $3+3+1+3+1=11$ ways to do so.

Those 11 ways can be formulated in the following manner:

1. $(a+c)(a-c)=(3\cdot5)^2(7^2\cdot13)^2=15^2\cdot637^2=202997^2-202772^2$ | $(a,c)=(202997, 202772)$ |

2. $(a+c)(a-c)=(3\cdot7)^2(5\cdot7\cdot13)^2=21^2\cdot455^2=103733^2-103292^2$ | $(a,c)=(103733, 103292)$ |

3. $(a+c)(a-c)=(3\cdot13)^2(5\cdot7^2)^2=39^2\cdot245^2=30773^2-29252^2$ | $(a,c)=(30773, 29252)$ |

4. $(a+c)(a-c)=(5\cdot7)^2(3\cdot7\cdot13)^2=35^2\cdot273^2=37877^2-36652^2$ | $(a,c)=(37877, 36652)$ |

5. $(a+c)(a-c)=(5\cdot13)^2(3\cdot7^2)^2=65^2\cdot147^2=12917^2-8692^2$ | $(a,c)=(12917, 8692)$ |

6. $(a+c)(a-c)=(7\cdot13)^2(3\cdot5\cdot7)^2=91^2\cdot105^2=9653^2-1372^2$ | $(a,c)=(9653, 1372)$ |

7. $(a+c)(a-c)=(3\cdot5\cdot13)^2(7^2)^2=195^2\cdot49^2=20213^2-17812^2$ | $(a,c)=(20213, 17812)$ |

8. $(a+c)(a-c)=(3\cdot5\cdot7\cdot7)^2(13)^2=735^2\cdot13^2=270197^2-270028^2$ | $(a,c)=(270197, 270028)$ |

9. $(a+c)(a-c)=(3\cdot5\cdot7\cdot13)^2(7)^2=1365^2\cdot7^2=931637^2-931588^2$ | $(a,c)=(931637, 931588)$ |

10. $(a+c)(a-c)=(3\cdot7\cdot7\cdot13)^2(5)^2=1911^2\cdot5^2=1825973^2-1825948^2$ | $(a,c)=(1825973, 1825948)$ |

11. $(a+c)(a-c)=(5\cdot7\cdot7\cdot13)^2(3)^2=3185^2\cdot3^2=5072117^2-5072108^2$ | $(a,c)=(5072117, 5072108)$ |

Edit:

$(a+c)(a-c)=(3\cdot5\cdot7\cdot7\cdot13)^2(1)^2=9555^2\cdot1^2=45649013^2-45649012^2$ and the last solution set to the problem would be $(a,c)=(45649013, 45649012)$!

Thank you

Last edited:

- Mar 31, 2013

- 1,335

So $9555^2 = 3^2 * 5^2 * 7^4 * 13^2$

This has (2+1)(2+1)(4+1)(2+1) or 135 factors out of which one is 9555 and 67 are below 9555 and 67 are above 9555

(a + c) > 9555 and (a-c) < 9555 is a set of solution

So number of solutions 67

For a we need to take$ 3^x * 5^ y * 7 ^ z * 13^ m$ (x,y,z,m to be chosen based on limit for example x between 0 and 3 such that a > 9555) and for c it is $9555^2/a$

- Thread starter
- #5

- Nov 29, 2013

- 172

Hello.

Thank you very much to all for taking part.

[tex]Let \ a, \ b, \ c, \ p, \ q \in{N} \ / \ a^2=b^2+c^2 \ and \ b=pq[/tex]

[tex]a=\dfrac{p^2+q^2}{2}[/tex]

[tex]b=pq[/tex]

[tex]c=\dfrac{p^2-q^2}{2}[/tex]

p | q | a | b | c |

1 | 9555 | 45649013 | 9555 | 45649012 |

3 | 3185 | 5072117 | 9555 | 5072108 |

5 | 1911 | 1825973 | 9555 | 1825948 |

7 | 1365 | 931637 | 9555 | 931588 |

13 | 735 | 270197 | 9555 | 270028 |

15 | 637 | 202997 | 9555 | 202772 |

21 | 455 | 103733 | 9555 | 103292 |

35 | 273 | 37877 | 9555 | 36652 |

39 | 245 | 30773 | 9555 | 29252 |

49 | 195 | 20213 | 9555 | 17812 |

65 | 147 | 12917 | 9555 | 8692 |

91 | 105 | 9653 | 9555 | 1372 |

Regards.

- Mar 31, 2013

- 1,335

above ans is wrong as it takes care of b = pq only and not c = pqHello.

Thank you very much to all for taking part.

[tex]Let \ a, \ b, \ c, \ p, \ q \in{N} \ / \ a^2=b^2+c^2 \ and \ b=pq[/tex]

[tex]a=\dfrac{p^2+q^2}{2}[/tex]

[tex]b=pq[/tex]

[tex]c=\dfrac{p^2-q^2}{2}[/tex]

p q a b c 1 9555 45649013 9555 45649012 3 3185 5072117 9555 5072108 5 1911 1825973 9555 1825948 7 1365 931637 9555 931588 13 735 270197 9555 270028 15 637 202997 9555 202772 21 455 103733 9555 103292 35 273 37877 9555 36652 39 245 30773 9555 29252 49 195 20213 9555 17812 65 147 12917 9555 8692 91 105 9653 9555 1372

Regards.

my ans is right

nt x = 9555* 9555;

#include <stdio.h>

main()

{

int a;

int b;

int i=0;

for ( a = 1; a < 9555; a ++)

{

b = x/a;

if( a * b == x) {

i++;

printf("the solution %i is %d %d\n", i, (a+b)/2, (b-a)/2);

}

}

}

the solution set is

the solution 1 is 45649013 45649012

the solution 2 is 15216339 15216336

the solution 3 is 9129805 9129800

the solution 4 is 6521291 6521284

the solution 5 is 5072117 5072108

the solution 6 is 3511469 3511456

the solution 7 is 3043275 3043260

the solution 8 is 2173773 2173752

the solution 9 is 1825973 1825948

the solution 10 is 1304275 1304240

the solution 11 is 1170507 1170468

the solution 12 is 1014445 1014400

the solution 13 is 931637 931588

the solution 14 is 724619 724556

the solution 15 is 702325 702260

the solution 16 is 608691 608616

the solution 17 is 501683 501592

the solution 18 is 434805 434700

the solution 19 is 390221 390104

the solution 20 is 310611 310464

the solution 21 is 270197 270028

the solution 22 is 260939 260764

the solution 23 is 234195 234000

the solution 24 is 202997 202772

the solution 25 is 186445 186200

the solution 26 is 167349 167076

the solution 27 is 145075 144760

the solution 28 is 140621 140296

the solution 29 is 133259 132916

the solution 30 is 103733 103292

the solution 31 is 100555 100100

the solution 32 is 90291 89784

the solution 33 is 87213 86688

the solution 34 is 78325 77740

the solution 35 is 71981 71344

the solution 36 is 62475 61740

the solution 37 is 56147 55328

the solution 38 is 54445 53600

the solution 39 is 47307 46332

the solution 40 is 44877 43848

the solution 41 is 39179 37996

the solution 42 is 37877 36652

the solution 43 is 34125 32760

the solution 44 is 30773 29252

the solution 45 is 29771 28196

the solution 46 is 27475 25760

the solution 47 is 24843 22932

the solution 48 is 21805 19600

the solution 49 is 21203 18928

the solution 50 is 20213 17812

the solution 51 is 19275 16740

the solution 52 is 17069 14144

the solution 53 is 16331 13244

the solution 54 is 15925 12740

the solution 55 is 14637 11088

the solution 56 is 14259 10584

the solution 57 is 13195 9100

the solution 58 is 12917 8692

the solution 59 is 12467 8008

the solution 60 is 11445 6300

the solution 61 is 10829 5096

the solution 62 is 10675 4760

the solution 63 is 10101 3276

the solution 64 is 9939 2736

the solution 65 is 9805 2200

the solution 66 is 9653 1372

the solution 67 is 9611 1036

- Thread starter
- #7

- Nov 29, 2013

- 172

It is not correct. You have had in account that ...?:above ans is wrong as it takes care of b = pq only and not c = pq

my ans is right

nt x = 9555* 9555;

#include <stdio.h>

main()

{

int a;

int b;

int i=0;

for ( a = 1; a < 9555; a ++)

{

b = x/a;

if( a * b == x) {

i++;

printf("the solution %i is %d %d\n", i, (a+b)/2, (b-a)/2);

}

}

}

the solution set is

the solution 1 is 45649013 45649012

the solution 2 is 15216339 15216336

the solution 3 is 9129805 9129800

the solution 4 is 6521291 6521284

the solution 5 is 5072117 5072108

the solution 6 is 3511469 3511456

the solution 7 is 3043275 3043260

the solution 8 is 2173773 2173752

the solution 9 is 1825973 1825948

the solution 10 is 1304275 1304240

the solution 11 is 1170507 1170468

the solution 12 is 1014445 1014400

the solution 13 is 931637 931588

the solution 14 is 724619 724556

the solution 15 is 702325 702260

the solution 16 is 608691 608616

the solution 17 is 501683 501592

the solution 18 is 434805 434700

the solution 19 is 390221 390104

the solution 20 is 310611 310464

the solution 21 is 270197 270028

the solution 22 is 260939 260764

the solution 23 is 234195 234000

the solution 24 is 202997 202772

the solution 25 is 186445 186200

the solution 26 is 167349 167076

the solution 27 is 145075 144760

the solution 28 is 140621 140296

the solution 29 is 133259 132916

the solution 30 is 103733 103292

the solution 31 is 100555 100100

the solution 32 is 90291 89784

the solution 33 is 87213 86688

the solution 34 is 78325 77740

the solution 35 is 71981 71344

the solution 36 is 62475 61740

the solution 37 is 56147 55328

the solution 38 is 54445 53600

the solution 39 is 47307 46332

the solution 40 is 44877 43848

the solution 41 is 39179 37996

the solution 42 is 37877 36652

the solution 43 is 34125 32760

the solution 44 is 30773 29252

the solution 45 is 29771 28196

the solution 46 is 27475 25760

the solution 47 is 24843 22932

the solution 48 is 21805 19600

the solution 49 is 21203 18928

the solution 50 is 20213 17812

the solution 51 is 19275 16740

the solution 52 is 17069 14144

the solution 53 is 16331 13244

the solution 54 is 15925 12740

the solution 55 is 14637 11088

the solution 56 is 14259 10584

the solution 57 is 13195 9100

the solution 58 is 12917 8692

the solution 59 is 12467 8008

the solution 60 is 11445 6300

the solution 61 is 10829 5096

the solution 62 is 10675 4760

the solution 63 is 10101 3276

the solution 64 is 9939 2736

the solution 65 is 9805 2200

the solution 66 is 9653 1372

the solution 67 is 9611 1036

[tex]a=\dfrac{p^2+q^2}{2}[/tex]

Regards.

- Mar 22, 2013

- 573

It is correct. $(a, c) = (9805, 2200)$ cannot be seen in either your or anemone's solution.

- Mar 31, 2013

- 1,335

I would like to see with evidence why it is not correctIt is not correct. You have had in account that ...?:

[tex]a=\dfrac{p^2+q^2}{2}[/tex]

Regards.

- Thread starter
- #10

- Nov 29, 2013

- 172

Sorry.above ans is wrong as it takes care of b = pq only and not c = pq

my ans is right

nt x = 9555* 9555;

#include <stdio.h>

main()

{

int a;

int b;

int i=0;

for ( a = 1; a < 9555; a ++)

{

b = x/a;

if( a * b == x) {

i++;

printf("the solution %i is %d %d\n", i, (a+b)/2, (b-a)/2);

}

}

}

the solution set is

the solution 1 is 45649013 45649012

the solution 2 is 15216339 15216336

the solution 3 is 9129805 9129800

the solution 4 is 6521291 6521284

the solution 5 is 5072117 5072108

the solution 6 is 3511469 3511456

the solution 7 is 3043275 3043260

the solution 8 is 2173773 2173752

the solution 9 is 1825973 1825948

the solution 10 is 1304275 1304240

the solution 11 is 1170507 1170468

the solution 12 is 1014445 1014400

the solution 13 is 931637 931588

the solution 14 is 724619 724556

the solution 15 is 702325 702260

the solution 16 is 608691 608616

the solution 17 is 501683 501592

the solution 18 is 434805 434700

the solution 19 is 390221 390104

the solution 20 is 310611 310464

the solution 21 is 270197 270028

the solution 22 is 260939 260764

the solution 23 is 234195 234000

the solution 24 is 202997 202772

the solution 25 is 186445 186200

the solution 26 is 167349 167076

the solution 27 is 145075 144760

the solution 28 is 140621 140296

the solution 29 is 133259 132916

the solution 30 is 103733 103292

the solution 31 is 100555 100100

the solution 32 is 90291 89784

the solution 33 is 87213 86688

the solution 34 is 78325 77740

the solution 35 is 71981 71344

the solution 36 is 62475 61740

the solution 37 is 56147 55328

the solution 38 is 54445 53600

the solution 39 is 47307 46332

the solution 40 is 44877 43848

the solution 41 is 39179 37996

the solution 42 is 37877 36652

the solution 43 is 34125 32760

the solution 44 is 30773 29252

the solution 45 is 29771 28196

the solution 46 is 27475 25760

the solution 47 is 24843 22932

the solution 48 is 21805 19600

the solution 49 is 21203 18928

the solution 50 is 20213 17812

the solution 51 is 19275 16740

the solution 52 is 17069 14144

the solution 53 is 16331 13244

the solution 54 is 15925 12740

the solution 55 is 14637 11088

the solution 56 is 14259 10584

the solution 57 is 13195 9100

the solution 58 is 12917 8692

the solution 59 is 12467 8008

the solution 60 is 11445 6300

the solution 61 is 10829 5096

the solution 62 is 10675 4760

the solution 63 is 10101 3276

the solution 64 is 9939 2736

the solution 65 is 9805 2200

the solution 66 is 9653 1372

the solution 67 is 9611 1036

You two have the reason, there are absent Pythagorean compound numbers.

A restriction is absent in the terms of reference of the question, that asks for " primitive solutions ". And nonetheless, they would exceed 4 of the solutions that are compound.(4ª, 7ª, 8ª, 12ª)

Thousand excuses.

Regards.