# 1/a + 1/b + 1/c = 1

## Main Question or Discussion Point

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

Related Linear and Abstract Algebra News on Phys.org
D H
Staff Emeritus

D H, I thought I was in the right forum. What I did so far: b+c = (a-1)*t and b*c = a*t and (c-1)*(b-1) = t+1. I'm stuck because I get too many posibilities.

CRGreathouse
Homework Helper
You can start by assuming, for the time being, that $a\le b\le c$ -- you can rearrange the terms later if you need to.

If a = 1 then the sum is too large, so $2\le a\le b\le c.$ If a = 2, $1/b+1/c=1/2$ 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).

D H
Staff Emeritus
You can start by assuming, for the time being, that $a\le b\le c$ -- you can rearrange the terms later if you need to.

If a = 1 then the sum is too large
That is true only if a, b, c are restricted to the positive integers. The OP just said integers.

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

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

Last edited:
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?

Last edited:
CRGreathouse
Homework Helper
Do you think there are more?
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.

Yes CRGreathouse, that's why I said that only one can be negative. My question was are there other solutions when one number is negative besides (t,1,-t)?

CRGreathouse
Homework Helper
If exactly one value is negative and another value is 1, the values are (1, v, -v). Otherwise, the sum of the reciprocals of the positive values is at most 1, so not a solution (since the negative will lower the result).

Need to find all the posible solutions (a,b,c) for 1/a + 1/b + 1/c = 1, a,b,c integers.
try to prove if there is finite amount of solutions

This is easy:
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?

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

I thought of it as well..
Just needed a second opinion.
Thanks! :)