Large Integer

What is the largest integer E, such that 1/A+1/B+1/C+1/D+1/E=1 ??
A<B<C<D<E are all positive integers

There is no such integer. As E approaches infinity, 1/E approaches zero.

Why does that prove there's no solution?

Actually, there is at least one solution:

1/3 + 1/4 + 1/5 + 1/6 + 1/20 = 1

This must be the greatest E (= 20) when A = 3, since 1/3 + 1/4 + 1/5 + 1/6 is the highest possible 4-term sum we can get starting at 1/3, making the last term (1/20) the smallest possible last term (i.e., the greatest value for the denominator).

There are no solutions for A > 3 since 1/4 + 1/5 + 1/6 + 1/7 + 1/8 < 1, and so it's impossible to reach a sum of 1 if the first term is 1/4 or smaller.

The only question remaining is whether it's possible to start with A=2 and find a combination that yields a higher value for E. I'll let someone else take it from here, as I'm very tired at the moment... ;-)

1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 1

(Sum of reciprocals of factors, excluding one, of a perfect number is 1, so I looked for one with the right number of factors)

No idea if it's the biggest E or not, though,

Originally posted by mtthw
1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 1

(Sum of reciprocals of factors, excluding one, of a perfect number is 1, so I looked for one with the right number of factors)

No idea if it's the biggest E or not, though,

I found 1/2+1/3+1/7+1/43+1/1806.

My idea was to start try to get the largest sum possible such that what remains is the lowest possible 1/x at each step but still > 0. This means I get the remainder and simply add 1 to its denominator.

First 1/2 => 1- 1/2 = 1/2
Second => 1/2-1/3 = 1/6
Third => 1/6-1/7 = 1/42
Fourth => 1/42-1/43 = 1/1806

So E is 1806. I don't think this can be bettered but I'm not sure I've proven it.

Originally posted by Palynka
I found 1/2+1/3+1/7+1/43+1/1806.

My idea was to start try to get the largest sum possible such that what remains is the lowest possible 1/x at each step but still > 0. This means I get the remainder and simply add 1 to its denominator.

First 1/2 => 1- 1/2 = 1/2
Second => 1/2-1/3 = 1/6
Third => 1/6-1/7 = 1/42
Fourth => 1/42-1/43 = 1/1806

So E is 1806. I don't think this can be bettered but I'm not sure I've proven it.

An exhaustive search shows you are right Palyanka, nice one! It would be interesting to prove it without a search though...

Originally posted by iamatiger
An exhaustive search shows you are right Palyanka, nice one! It would be interesting to prove it without a search though...

Interesting would be if you can rewrite it to a formula.
What number would 'n' be when 1/A + 1/B + ... 1/n = 1?
Where A, B, ... n are all positive integers.

First Lets make a difference between an 'ending integer' and an 'integer in line'
'ending integer' = 'integer in line' - 1
Below I will only work with 'integers in line'

A=2
B=3
C=7
D=43
This seems like a function line!
B= A+1
C= B*A+1
D= C*B*A+1
Lets see if this works for E
E= 2*3*7*43+1 = 1807

'ending integer' = 'integer in line' - 1
1807 - 1 = 1806

Continuing
F= 2*3*7*43*1807+1 = 3263443
and the ending number 3263442 is indeed the right answer to F
G would be 2*3*7*43*1807*326443+1 and so on

This is where my math ends. I learned how this can be made into a formula. Unfortunately my memory fails. Also it is 2 AM here and I CBA to solve this.

Originally posted by yashin
Interesting would be if you can rewrite it to a formula.
What number would 'n' be when 1/A + 1/B + ... 1/n = 1?
Where A, B, ... n are all positive integers.

First Lets make a difference between an 'ending integer' and an 'integer in line'
'ending integer' = 'integer in line' - 1
Below I will only work with 'integers in line'

A=2
B=3
C=7
D=43
Th ...[text shortened]... la. Unfortunately my memory fails. Also it is 2 AM here and I CBA to solve this.

Cool, now we just need to prove that a line designed by that formula is optimal. It must be something like showing that this line grows the denominators at the maximum rate possible.

Originally posted by yashin
Interesting would be if you can rewrite it to a formula.
What number would 'n' be when 1/A + 1/B + ... 1/n = 1?
Where A, B, ... n are all positive integers.

First Lets make a difference between an 'ending integer' and an 'integer in line'
'ending integer' = 'integer in line' - 1
Below I will only work with 'integers in line'

A=2
B=3
C=7
D=43
Th la. Unfortunately my memory fails. Also it is 2 AM here and I CBA to solve this.

Err... Isn't that what I said? The remainder is always 1/(A*B*...*(m-1)) at each step m.

Originally posted by Palynka
Err... Isn't that what I said? The remainder is always 1/(A*B*...*(m-1)) at each step m.

Yes you did more or less, but you can see by iamatiger's reaction he didn't understand the method you used. It seemed like you found the number where ... + 1/m =1 and then added 1 to m. It wasn't clear to me how you got the answer, but since you were first the credit is all yours 🙂.

Originally posted by yashin
Yes you did more or less, but you can see by iamatiger's reaction he didn't understand the method you used. It seemed like you found the number where ... + 1/m =1 and then added 1 to m. It wasn't clear to me how you got the answer, but since you were first the credit is all yours 🙂.

iamatiger is a pretty smart guy, I'd give him more credit. But it's true that you explained it much more clearly. 🙂

My idea was first to start with the largest 4 element sum possible for integers larger than one (1/2+1/3+1/4+1/5) and work backwards. The idea is that E will be the largest the closer I get to 1. But as I was doing it I realized that at every step of the sum the remainder was of the form 1/x and so if I subtract the largest possible number _ 1/(x+1) _ that still leaves the sum < 1 then the remainder is also of the same form (namely _ 1/[x*(x+1)] _ ). This is how I got there, with a bit of luck and intuition. I'm usually terrible with puzzles that deal with integers only.

no response from thudnblunder, but it seems you are right.

Yes, Palynka's solution is correct. Well done!

Originally posted by ThudnBlunder
Yes, Palynka's solution is correct. Well done!

Thanks!

Do you have a definite proof?