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Square

cinelli
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Square

#303202

Postby cinelli » April 25th, 2020, 2:24 pm

A mathematical problem. Find all values of integer n > 1 for which the sum

25 + 38 + 41 + ... + (8n+17)

is a perfect square.

Cinelli

UncleEbenezer
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Re: Square

#303214

Postby UncleEbenezer » April 25th, 2020, 3:33 pm

That looks more like a problem in lateral thinking than anything mathematical. As in, what is your ellipsis?

In the absence of ellipsis, it just evaluates as 121 + 8n being a square, which is a rather dull problem: for every odd x > 11, let y = (x-11) / 2. Then x2 - 121 = (x-11) (x+11) = 2y * (2y+22) = 4y(y+11), which is always divisible by 8. So the values of n are the square roots of every (x2-121) for odd values of x starting at 13. That gives us n=6,13,21,30, ...

I haven't bothered with a spoiler 'cos, being one of yours, I'm sure it's not supposed to be that straightforward. :o

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Re: Square

#303237

Postby GoSeigen » April 25th, 2020, 5:54 pm

cinelli wrote:A mathematical problem. Find all values of integer n > 1 for which the sum

25 + 38 + 41 + ... + (8n+17)

is a perfect square.

Cinelli


Presumably a small typo there, should read:


A mathematical problem. Find all values of integer n > 1 for which the sum

25 + 33 + 41 + ... + (8n+17)

is a perfect square.

UncleEbenezer
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Re: Square

#303255

Postby UncleEbenezer » April 25th, 2020, 6:52 pm

GoSeigen wrote:A mathematical problem. Find all values of integer n > 1 for which the sum

25 + 33 + 41 + ... + (8n+17)

is a perfect square.

That at least makes a more obvious problem. Your series reduces to 17*n + 8 * n * (n+1) /2, or more simply n(4n+21). There's one obvious solution to that, where both n and (4n+21) are squares. But I don't instantly see a proof of uniqueness, and both factors being squares is a stronger condition than you've asked.

cinelli
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Re: Square

#303267

Postby cinelli » April 25th, 2020, 8:44 pm

GoSeigen wrote:
Presumably a small typo there, should read:

25 + 33 + 41 + ... + (8n+17)

is a perfect square.


You are right - it should be 33. Apologies.

Cinelli

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Re: Square

#303291

Postby modellingman » April 26th, 2020, 1:50 am

Spoiler







As noted by Uncle E
25 + 33 + 41 +... + (8n + 17) = n(4n + 21)

A bit of brute force reveals the first 2 solutions occur at n=1 (sum is 25) and at 25 (sum is 3025, square of 55). So is there a pattern here? Trying n as 3025, yields a sum of 36,666,025, but this is not a perfect square. So no pattern based on generating a new solution from the preceding one.

However, looking at the next highest square number for n in the range 4 to 25 does reveal something useful. The next highest square greater than or equal to n(4n + 21) is always the square of s = 2n+5.

(2n + 5)(2n + 5) - n(4n + 21) = 25 - n

which is positive for n less than 24, 0 when n is 25 and negative for n greater than 25.

It turns out that the square of s + 1 is always greater than n(4n + 21) since

(s + 1)(s + 1) - n(4n + 21) = 36 + 3n

Since

s^2 < n(4n + 21) < (s + 1)^2

for all n > 25 then for this same range, n(4n + 21) cannot be a perfect square. Therefore, since there are two solutions for n<=25, these must be the only solutions.

cinelli
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Re: Square

#303561

Postby cinelli » April 27th, 2020, 11:53 am

Reply to spoiler:

Well solved modellingman. It is satisfying when two different methods produce the same answer. I will give my method in a couple of days.


Cinelli

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Re: Square

#304059

Postby UncleEbenezer » April 29th, 2020, 10:26 am

cinelli wrote: I will give my method in a couple of days.

Cinelli


Is your method to enumerate candidate square roots?

The square root of sq = 4n2 + 21n is something above 2n, let's say 2n+x. There are few enough cases to enumerate:

For x <= 2, (2n+x)2 <= 4n2 + 8n + 4 < sq
For x >= 6, (2n+x)2 >= 4n2 + 24n + 36 > sq

That leaves us cases 3, 4 and 5 as candidates.
3: 4n2 + 12n + 9
4: 4n2 + 16n + 16
5: 4n2 + 20n + 25

But we can reduce this to trivial linear equations by subtracting 4n2 from the squares and from sq:
12n + 9 = 21n has solution 1
16n + 16 = 21n has non-integer solution
20n + 25 = 21n has solution 25

So 1 and 25 are indeed the only solutions.

cinelli
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Re: Square

#304220

Postby cinelli » April 29th, 2020, 6:11 pm

Here is my solution:

Sum of the AP is n*(4n+21) so we are looking for positive integer k for which

4n^2+21*n = k^2

Multiply through by 16 to avoid fractions: 64*n^2 + 16*21*n - 16*k^2 = 0

Complete square: (8n+21)^2 – 441 – 16*k^2=0

Factorise difference of two squares: ((8n+21)+4k) * ((8n+21)-4k) = 21^2

Possible factorisations of RHS are 441,1 147,3 49,9 63,7 to give two simultaneous equations for n and k. Solving these gives

n=25 k=55
n=27/4 k=18
n=1 k=5
n=7/4 k=4

So just 2 solutions for integers n and k: n=1 and 25. Two series are 25 = 5^2 (trivial) and 25 + 33 + … + 217 = 3025 = 55^2

Cinelli

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Re: Square

#304245

Postby modellingman » April 29th, 2020, 7:32 pm

OK, now we have had 3 solutions, what about these variations on Cinelli's series?

33 + 41 + 48 + ... + (8n + 17) - first term dropped
17 + 25 + 33 + ... + (8n + 17) - new first term inserted
26 + 34 + 42 + ... + (8n + 18) - first term increased by 1

modellingman

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Re: Square

#304286

Postby UncleEbenezer » April 29th, 2020, 9:54 pm

modellingman wrote:OK, now we have had 3 solutions, what about these variations on Cinelli's series?

33 + 41 + 48 + ... + (8n + 17) - first term dropped
17 + 25 + 33 + ... + (8n + 17) - new first term inserted
26 + 34 + 42 + ... + (8n + 18) - first term increased by 1

modellingman

Those yield to exactly the same approach I already used. The solutions are respectively {50}, {8} and {}.

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Re: Square

#304453

Postby modellingman » April 30th, 2020, 12:01 pm

UncleEbenezer wrote:
modellingman wrote:OK, now we have had 3 solutions, what about these variations on Cinelli's series?

33 + 41 + 48 + ... + (8n + 17) - first term dropped
17 + 25 + 33 + ... + (8n + 17) - new first term inserted
26 + 34 + 42 + ... + (8n + 18) - first term increased by 1

modellingman

Those yield to exactly the same approach I already used. The solutions are respectively {50}, {8} and {}.


Indeed. And you spotted the inconsistency I introduced with n. I should, of course, have written

33 + 41 + 48 + ... + (8n + 25)
17 + 25 + 33 + ... + (8n + 9)

which adjusts your results to {49}, {9} and {}.

modellingman


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