- #1

- 26

- 0

Need to find all the posible solutions (a,b,c) for 1/a + 1/b + 1/c = 1, a,b,c integers.

- Thread starter xax
- Start date

- #1

- 26

- 0

Need to find all the posible solutions (a,b,c) for 1/a + 1/b + 1/c = 1, a,b,c integers.

- #2

- 15,393

- 685

xax, please show your work, and post your homework questions in the correct forum.

- #3

- 26

- 0

- #4

CRGreathouse

Science Advisor

Homework Helper

- 2,820

- 0

If a = 1 then the sum is too large, so [itex]2\le a\le b\le c.[/itex] If a = 2, [itex]1/b+1/c=1/2[/itex] so clearly b needs to be no more than 4 (to get 1/4 + 1/4 = 1/2). Check the cases of b = 3 and b = 4.

If a = 3 then b and c need to be small to make the sum 1, so check cases similarly.

Finally, rearrange the solutions you have as needed (so a need not be the smallest).

- #5

- 15,393

- 685

That is true only if a, b, c are restricted to the positive integers. The OP just said integers.You can start by assuming, for the time being, that [itex]a\le b\le c[/itex] -- you can rearrange the terms later if you need to.

If a = 1 then the sum is too large

- #6

CRGreathouse

Science Advisor

Homework Helper

- 2,820

- 0

True, I missed that. There is one family of solutions which includes negative numbers.That is true only if a, b, c are restricted to the positive integers. The OP just said integers.

- #7

- 188

- 1

for example if a=-b an c is arbitrary big or similar we have an approximate solution in integers to your equation.

or simply a=-b and c=1 b=-c and a=1 a=-c and b=1

or simply a=-b and c=1 b=-c and a=1 a=-c and b=1

Last edited:

- #8

- 26

- 0

Thanks alot to all of you and expecialy to CRGreathouse. I've proven that a<=3, b<=4 and found all the solutions(there are 3 in total).

Edit: If one of them is negative the solution is (t,1,-t). Do you think there are more?

Edit: If one of them is negative the solution is (t,1,-t). Do you think there are more?

Last edited:

- #9

CRGreathouse

Science Advisor

Homework Helper

- 2,820

- 0

If two are negative you have 1 - 1/a - 1/b < 1. If three are negative you have -1/a - 1/b - 1/c < 0.Do you think there are more?

- #10

- 26

- 0

- #11

CRGreathouse

Science Advisor

Homework Helper

- 2,820

- 0

- #12

- #13

- 2

- 0

2, 3, 6

2, 6, 3

3, 2, 6

3, 6, 2

6, 2, 3

6, 3, 2

6 solutions..

1/2 + 1/3 + 1/6 = 1.

Then i got a new question.

1/a + 1/b + 1/c = 7.

If a, b, c, are integers, how many solutions are there?

- #14

- 907

- 2

Zero! Because 1/a, 1/b and 1/c are all <=1. Therefore their sum is <=3.

- #15

- 2

- 0

I thought of it as well..

Just needed a second opinion.

Thanks! :)

Just needed a second opinion.

Thanks! :)

- Replies
- 2

- Views
- 3K

- Replies
- 7

- Views
- 857

- Replies
- 2

- Views
- 926

- Last Post

- Replies
- 1

- Views
- 2K

- Replies
- 14

- Views
- 946

- Replies
- 2

- Views
- 655

- Replies
- 12

- Views
- 888

- Replies
- 2

- Views
- 2K

- Replies
- 2

- Views
- 4K

- Replies
- 4

- Views
- 812