Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: How to solve these equations

Hi mr.wong;

I followed the discussion but kept my mouth shut because although ax-by=c has methods to solve it such as Brahmagupta's method I was not sure whether your other conditions would affect the method. Anyway, zetafunc seems to have answered that for me and I can again keep my mouth shut.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: How to solve these equations

If we know explicitly the set of possible (i,j) without the imposed conditions, then imposing simple conditions can be quite easily dealt with (you might want i and j to be coprime, you might want i to be a square and j to be a cube, etc -- there are techniques to deal with these). But it can get very difficult if you impose trickier conditions on your i,j.

I would strongly recommend learning about modular arithmetic so you can easily deal with equations like these -- a book on elementary number theory would be useful (I can also provide lecture notes). You might be surprised at the kinds of things number theory can solve these days!

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: How to solve these equations

Hi  zetafunc ,

Will  you  please  give  an  example  ? 

Hi  thickhead ,

Once  you  have  got  a  valid  pair  of  i  and  j , then  you  may  add 
n * 182 * 165  to  both  sides  of  182 * i   and  165 * j  and  obtain  a  bunch  of  i  and  j   as  you  had  stated  in  # 3 . While  to  express  a  certain  prime   as   a * 17 +  b * 5   may  need  trial  and  error .

That trial and error is quite easy because of 5.e.g. 173 You have to find a multiple of 17 ending with 3 or 8 i.e 68+105 or 153+20  but the resulting i and j you have to check for the condition.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

mr.wong

Member

Registered: 2015-12-01

Posts: 252

Re: How to solve these equations

Hi  zetafunc  ,

Thanks  for  your  recommendation  .  However  ,  for 
my  age  now  it  is  not  easy  to  learn  new  topics  in 
mathematics .

Hi  thickhead ,

You  are  right  and  it  seems  the  resulting  i  will 
naturally  be  coprime  with  165  and  j  , while  the 
resulting  j  will  naturally  be  coprime  with  182  and  i .

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: How to solve these equations

Well, here's a short article you can read about which might help, if you are interested: https://en.wikipedia.org/wiki/Modular_a … e_relation