Let $n$ be a fixed natural number. How to solve the following equation in natural numbers: $$ \frac{1}{x_1} + \frac{2}{x_2} + \cdots + \frac{n}{x_n} = 1 $$ (I can find many soltions but I am looking for all solutions)
Asked
Active
Viewed 182 times
3
-
What are $x_1...$ integers or natural numbers – Archis Welankar Jan 05 '16 at 05:21
-
@ArchisWelankar natural numbers – alex alexeq Jan 05 '16 at 05:21
-
It is sufficient to solve the equation. $$\frac{1}{x_1}+\frac{2}{x_2}+\frac{3}{x_3}=\frac{a}{b}$$ Number $a,b - $ will be set. Which are obtained if we ask the other numbers at their discretion. – individ Jan 05 '16 at 05:32
-
You can use this formula. http://math.stackexchange.com/questions/450280/erdös-straus-conjecture/831870#831870 – individ Jan 05 '16 at 05:44
-
I see what you mean that it's trivial to write down some solutions e.g. $x_i=in$. But do you have a good reason to believe it's easy to characterize all solutions? That seems like it may be intractable. – Gregory Grant Jan 05 '16 at 06:06
-
@GregoryGrant I have no reason to believe that this is an easy queastion, it may be even an open question. – alex alexeq Jan 05 '16 at 06:11
-
Well it's easy if $n=2$. The condition $x_2-2\mid x_2$ means $x_2$ can't be very big, it would have to be no bigger than $4$. – Gregory Grant Jan 05 '16 at 06:21
-
@GregoryGrant It would be easier for n=1. – alex alexeq Jan 05 '16 at 06:26
-
Well I was working on $n=3$ and making some headway, but the sarcasm I can do without, so I think I'll look elsewhere for my late night entertainment. – Gregory Grant Jan 05 '16 at 06:57
1 Answers
0
We assume x1 < x2/2 < x3/3 ... and We may write 1/(x1/1) +1/(x2/2) +1/(x3/3) +. . . 1/(xn/n)=1 Number of terms is n then 1/(x1/1)<1/n. taking x1 an arbitrary value satisfying 1/(x1) < 1/n and finding equation:
1/(x2/2) +1/(x3/3) +. . . 1(xn/n)=1-1/x1
Now 1/(x2/2) < 1-1/x1 defines lowest value and also n-1/(x2/2) > 1-1/x1 defines upper value of x2. we take a value in this range. In this way we can find xi; i=3, 4, 5 ... .
Example: 1/2 + 2/6 + 3/21 + 4/172 + 5/9030 = 1
sirous
- 12,694