Problem 135

[Marcscol]

Hi Euler,

I was trying to solve problem 135 at it says that the least value that an equation has exactly two solutions is 27:

34^2 - 27^2 - 20^2 = 12^2 - 9^2 - 6^2 = 27

Trying to solve it I got that 15 has two solutions also:

19^2 - 15^2 - 11^2 = 7^2 - 5^2 - 3^2 = 15

Maybe I'm wrong, maybe I'm doing something wrong, but this is the thing that I've noticed.

Thanks and Regards,

Marcus

[xenon]

How about 5^2 -3^2 -1^2?

[Marcscol]

Yeah.... After posting, I checked the algo and I saw that I was making a mistake.
anyway, thanks for the reply.

Marcus

[GoSlow2GoFast]

I've been looking at problem 135, and seem to be missing something. It states that for n=27, there are only 2 solutions:

34² - 27² - 20² = 27
12² - 9² - 6² = 27

Why aren't others included, such as:

8² - 6² - 1² = 27
10² - 8² - 3² = 27
14² - 12² - 5² = 27
16² - 15² - 2² = 27

I must be missing something in the wording of this problem, but it's not jumping out at me.

~gs2gf

[quilan]

The numbers have to be part of an arithmetic progression.

i.e. (x+2k)^2 - (x+k)^2 - x^2 = n

[GoSlow2GoFast]

Duh, I was completely "reading over" that clause and it just wasn't being noticed. Thanks.

~gs2gf

[MaJJ]

If I use quilan's words -

Code: Select all

(x+2k)^2 - (x+k)^2 - x^2 = n

- then my loop stops when k = 578 and

Expand

605 values of n have 10 solutions

. Is the latter way off?

The k gives sense:
(1+578+578)^2 - (1+578)^2 - 1^2 = 1338649 - 335241 - 1 = 1003407
(1+577+577)^2 - (1+577)^2 - 1^2 = 1334025 - 334084 - 1 = 999940
- or am I missing something?

[TripleM]

I'm afraid your answer isn't close.

[kapila_619]

I don't know whether this is a valid question
I think I'm getting my maths wrong somewhere

to find, values of z for which (z+2k)^2-(z+k)^2-z^2=n

f(z,k)=z^2-2zk-3k^2+n //n is const
I have found condition for z>0 to be : k belongs to -sqrt(n)/2 to sqrt(n)/2

but
n=27,x=7,y=20
and
n=27,x=3,y=6

these don't satisfy my obtained conditioned, while they satisfy f(z,k)=0

for n=27, k should belong to -2.598 to 2.598
but k=3,7 satisfy

I know we have to find f(z,k)=0, but the problem says x,y,z>0, so i just applied the condition discriminant<0 to get a range of values of 'k' for a particular 'n'

can u tell me where am I going wrong

thanks

[jaap]

I have found condition for z>0 to be : k belongs to -sqrt(n)/2 to sqrt(n)/2

I think you have a sign error on the k^2 term of the discriminant. The one I get is always positive (for positive n) so of no use.

Edit: No, I got the sign wrong myself. So you get an expression in k and n for the discriminant. You need that discriminant to be positive for there to be any solutions for z at all. So you need k to lie outside the region you found rather than inside.

[lirugeda]

I found an algorithm to get the answer, but when I put my answer in, it's incorrect. So, I was wondering if someone could help me verify the first 10 values of n which have exactly 10 distinct solutions:

1155
1995
2415
3003
3135
3255
3315
3795
3927
4095

[jaap]

The correct ones are 1155, 3003, 3135 (which are first, 7th and 8th on my list).

[xe3tec]

are these the first 3?

[spoiler]1155
1184
1203[/spoiler]

[jaap]

No. The first one is correct, but the others don't have 10 solutions but only two.

[xe3tec]

edit: ok thx

I am stuck

[xe3tec]

[spoiler]1155
[1155, 1575, 1755, 1920, 2079, 2304, 2475, 2688, 2835, 2880, 3003][/spoiler]

which one of these are wrong?

[Tharsis]

The correct ones are 1155, 3003, 3135 (which are first, 7th and 8th on my list).

Actually, 3315 is also correct.

[spoiler]1155
[1155, 1575, 1755, 1920, 2079, 2304, 2475, 2688, 2835, 2880, 3003][/spoiler]

which one of these are wrong?

1575, 2475, 2835, and 2880 are not correct.

[thkang]

Given the positive integers, x, y, and z, are consecutive terms of an arithmetic progression, the least value of the positive integer, n, for which the equation, x^2 -y^2- z^2 = n, has exactly two solutions is n = 27:

34^2- 27^2- 20^2 = 12^2- 9^2-6^2 = 27

It turns out that n = 1155 is the least value which has exactly ten solutions.

How many values of n less than one million have exactly ten distinct solutions?

I'm having problem with counting N:

let's say we're thinking about arithmetic progression that progresses by 10. since 0<n<1,000,000, here are all the possible cases:

Code: Select all

P||||    X   Y  Z  N
p==10 21 11 1 319
p==10 22 12 2 336
p==10 23 13 3 351
p==10 24 14 4 364
p==10 25 15 5 375
p==10 26 16 6 384
p==10 27 17 7 391
p==10 28 18 8 396
p==10 29 19 9 399
p==10 30 20 10 400
p==10 31 21 11 399
p==10 32 22 12 396
p==10 33 23 13 391
p==10 34 24 14 384
p==10 35 25 15 375
p==10 36 26 16 364
p==10 37 27 17 351
p==10 38 28 18 336
p==10 39 29 19 319
p==10 40 30 20 300
p==10 41 31 21 279
p==10 42 32 22 256
p==10 43 33 23 231
p==10 44 34 24 204
p==10 45 35 25 175
p==10 46 36 26 144
p==10 47 37 27 111
p==10 48 38 28 76
p==10 49 39 29 39

now the problem is I'm having hard time understanding what n I should count.
the least value of the positive integer, n,
I can see that 49, 39, 29 yields least value of n, which is 39.
(and this is trivial without enumerating all x, y, z)

initially, given this observation, I thought what I had to do was:
for p==10, the least value of n is 39, count it once, move along to next p
however, by this method the given example of n=1155 having 10 solutions could not be found.


edit :
disregard everything I wrote.

however I have a question : how could n have a 'least' value? doesn't that term implies there is also a 'most' value and there are more than one possible 'n's for an arithmetic sequence?

[TheEvil]

however I have a question : how could n have a 'least' value? doesn't that term implies there is also a 'most' value and there are more than one possible 'n's for an arithmetic sequence?

If I remember well, in most of the cases least means the smallest number.
By this I mean: let us fix an arbitrary n which is less than 1155. The number of different triplets (x,y,z) for which x^2-y^2-z^2=n for this fixed n, is at most nine, or at least eleven. For n=1155, the number of triplets is exactly ten.

[JMW1994]

On this problem, is there a difference between "exactly ten solutions" and "exactly ten distinct solutions?" I found this very confusing since my code is able to get jaap's numbers in the correct order and yet my answer is wrong.