# Interesting equation with fractions

Find distinct positive integers $$a$$, $$b$$, $$c$$ and $$d$$ such that

$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{d} = \frac{1}{a} \cdot \frac{1}{b} \cdot \frac{1}{c} \cdot\left( -\frac{1}{d}\right)$$

No calculators or computers allowed.

• Associates to equations of theoretical physics.
– z100
Mar 3 at 22:14
• @z100 ooh please tell us more Mar 3 at 22:26
• @Dmitriy Kamenetsky left side reminds me of some Minkowski space equations (with a space-time as a model, where time coordinate has different sign to 3 space coordinates). BTW, OP's equation can be simplified to D-(A+B+C)=ABCD along the condition A,B,C,D are all inverse naturals.
– z100
Mar 4 at 19:59

These work

a=7, b=3, c=2, d= 1

Method used

First put both sides of the equation over a common denominator, namely abcd.

This gives

bcd+acd+abd-abc=-1

Notice that

if d is very large, then this will probably not work, so try d=1

This gives

bc+ac+ab-abc = -1

Try

c=2 to keep the numbers small

This gives

2b+2a+ab-2ab = -1

Try

b=3 to keep the numbers small

This gives

6+2a+3a-6a = -1

Solving gives

-a = -7 or a=7

There are a lot of solutions, e.g.

$$a=bcd+1,b=cd+1,c=d+1,d$$
$$a=bcd+1,b=(cd+1)/2,c=d+2,d$$ (for odd d)
$$a=bcd+1,b=(cd+1)/5,c=d+5,d$$ (for d 2mod5 or 3mod5)
examples:
43 7 3 2
119 8 5 3
1429 17 12 7