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Where were you when you realized this had...
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You are currently reading a thread in /sci/ - Science & Math

File: pyth4.png (1 KB, 307x110) Image search: [iqdb] [SauceNao] [Google]
1 KB, 307x110
Where were you when you realized this had no integer solutions, /sci/?
>>
>>7841263
>pls /sci/ do my homework
>>
>>7841263
(0,0,0,0)
>>
>>7841263
a = 2
b = 2
c = 1
g = 3

Troll or clever hw bait? You could probably write a program to find hundreds of integer solutions in seconds.
>>
[3,0,0,3]
[0,75,0,75]
[0,0,12,12]
>>
>>7841263
run this in python
for a in range(100):
for b in range (100):
for c in range (100):
for g in range(100):
if a**2 + b**2 + c**2 == g**2:
print(a,b,c,g)
There are a lot of solutions
>>
>>7841275
no solutions with odd integers
>>
>>7841281
Do you mean none of the integers are odd?
>>
>>7841281
Wow this is weird I just ran my previous program except with the parameters that each number must be odd. Nothing has shown up yet (at least in the range of 100). Is there a proof of why this is somewhere?
>>
>>7841301

Suppose you have two odd integers and multiply them, what can you say about the result?

Suppose you have three odd numbers and add them together, what can you say about the result?

Also: OP is an idiot.
>>
>>7841314
It's odd in both cases. Proves nothing.
>>
>>7841263
let a^2+b^2=d^2
then d^2+c^2=g^2
so it is doable
>>
>>7841263
if a,b,c and g are odd, then they are either equal to 1 or 3 mod 4

this means that a^2,b^2,c^2 and g^2 are equal to 1 mod 4

so a^2+b^2+c^2 is equal to 3 mod 4

and g^2 is equal to 1 mod 4

therefore the equality never holds.
>>
>>7841389
Sqrt odd = odd
>>
>>7841429
Once again, this proves nothing.
>>
>>7841416
Nice
>>
>>7841416
wrong
>>
>>7841496
because...?
>>
>>7841263
for a in range(10000):

for b in range(10000):

for c in range(10000):

for g in range(10000):

if (a*a + b*b + c*c == g*g):

print a,b,c,g

more than 1000 solutions dipshit
>>
>>7841263

The OP can be interpreted as a poor/incomplete phrasing of the problem of finding a /perfect cuboid/, which is tantamount to solving a system of /four/ diophantine equations, the three simpler of which OP has conveniently forgotten (these equations describe a box which also has integer /face diagonals/). Of course, it is also possible that OP was trolling with reference to pythagorean quadruples themselves, but as this poster >>7841275 has correctly observed, this is simple, and so OP does not deserve the benefit of the doubt with his intent.

The previous poster has therefore correctly described a box with three >>7841273 (almost) NON-TRIVIAL integer edge lengths, as well as one integer spatial diagonal. The more interesting unsolved curiosity of the perfect cuboid, however, requires seven integer lengths, being the above plus the face diagonals. Such an object has neither been shown to exist, nor disproven as of yet.

A simple treatiste on Pythagorean triangles is due to Sierpinski (Dover), and toward the end of this, he makes some remarks on the properties of Euler Bricks (integer edges and face diagonals but not the spatial diagonal, or 3/4 of the above diophantine equations, or 6/7 of the lengths of a perfect cuboid), which amount to restrictions on any existing perfect cuboid. Computer searches have been done up to 10^13 or somewhere in that neighborhood, finding no cuboid. I myself became interested in this problem due to /sci/, and I proved a few lemmas about Euler Bricks some months ago which I haven't seen elsewhere.

[cont]
>>
>>7841263
a = k, b =2k, c = 2k, g = 3k and k is any integer.
>>
There are no integer solutions where a, b, c, g are all different from each other
>>
>>7841263
Well no shit. "a b c g" are letters not numbers you dumbfuck
>>
File: keklel.png (79 KB, 1216x686) Image search: [iqdb] [SauceNao] [Google]
79 KB, 1216x686
[cont]

I began with a definition and some conventions, which are consistent with wiki's treatment (and not with mathworld's, which I dislike and find unnecessarily tedious). As I move along I find that there is always a short, middle and long edge, and a short, middle and long face diagonal, and a spatial diagonal, which I always call a,b,c,d,e,f and g, respectively, as wiki does. Here's the cute thing: /It may happen that c>d or vice verse/, pic related.

DEFINITION: An EULER BRICK is a cuboid which has three natural edge lengths, or edges, and three natural face diagonals, where "natural" excludes zero.

Lemma 1: No two edges are equal. (pf: pythag. thm & sqrt2 is irrational, RAA)
Corollary: There are always a short, middle and long edge.

Lemma 2: No Euler Brick has a unit edge. (pf: RAA, squares are sums of odd numbers)

Lemma 3: No two face diagonals are equal. Moreover, they correspond to the edges in the expected way, via the equations. (pf: use L1 and L2, RAA, and manipulate equalities and inequalities)
Corollary: There are always a short, middle and long face diagonal.

Lemma 4: /Depending on the Brick/, the long edge may be longer than the short face diagonal, /or vice verse/. (pf: inspect (240,252,275) and (44,117,240) )

Lemma 5: (Incomplete, pic related): a-g^2 satisfy a given chart of inequalities. (pf. so far: a long, sequential train of thought, starting with 1), 2) etc)

Conjecture: For any given Euler Brick, c-b =/= b-a. (I may have actually proven this using Almost-Isosceles right triangles, but I'd need to review my train of thought. Nyblom of Australia had a very helpful paper for as far as I took it)
>>
>>7841723

False.

2^2 + 3^2 + 6^2 = 49 = 7^2. Search the phrase "Pythagorean quadruple" for many other (counter)examples of this.
>>
>>7841743

Another larger conjecture went something like this: "There does not exist an Euler Brick whose edge lengths are in arithmetic progression."

Now this is jogging my memory; there's two or three cases one has to check for this. IIRC I did in fact establish that for all bricks c-b = b-a , which at first blush sounds like I took care of the above, but that's not how I remember it going. I'm probably phrasing something wrong at the moment.
>>
>>7841757
I rembember the brick discussions. Worked in it a bit. Got a little somewhere... Was thinking of writing it down and submitting to unsolved problems.org where similar stuff us posted. Then I saw the guy wants \$50 to post there! Not worth it.
>>
>>7841708
>>7841743
>>7841757
pure autism
>>
>>7841281
>>7841416
>>7841723
>>7841741
Autism Gone Wild
>>
>>7842326

Autists sometimes have the decency to have their remarks as factually correct, and they claim to care about facts; since most of your above is clearly false, it can't fairly be put down to autists as such.
>>
>>7842326
>>7842399
samefag
>>
File: 1451934097987.gif (2 MB, 235x240) Image search: [iqdb] [SauceNao] [Google]
2 MB, 235x240
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0 0 9489 9489
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0 0 9499 9499
0 0 9500 9500
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Muh solutions for range(1000)
Kill yourself faggot.
>>
>>7841263
Your mom's house. Which is actually your house too, except we weren't in the basement.
>>
>>7841482
Stop b8ing pls family
>>
Bruteforcing this problem in Python returns a lot of solutions, without repeating any:

for a in xrange(2, 1000, 2):
for b in xrange(a, 1000, 2):
for c in xrange(b, 1000, 2):
for g in xrange(c, 1000, 2):
if a**2 + b**2 + c**2 == g**2:
print(a,b,c,g)

The first ones found:

2 4 4 6
2 8 16 18
2 12 36 38
2 16 64 66
2 20 100 102
2 24 24 34
2 24 144 146
2 28 196 198
2 32 256 258
2 36 60 70
2 36 324 326
2 40 400 402
2 44 92 102
2 44 484 486
2 48 576 578
2 52 676 678
2 56 152 162
2 56 784 786
2 60 900 902
2 64 200 210
2 68 76 102
2 76 284 294
2 84 348 358
2 88 136 162
2 96 456 466
2 104 536 546
2 116 668 678
2 120 264 290
2 124 764 774
2 128 224 258
2 136 160 210
2 136 920 930
2 140 140 198
2 140 364 390
2 144 288 322
2 164 244 294
2 172 556 582
2 192 696 722
2 196 548 582
2 208 344 402
2 212 644 678
2 224 952 978
2 228 276 358
2 236 532 582
2 256 536 594
2 264 384 466
2 264 672 722
2 272 296 402
2 284 508 582
2 308 604 678
2 324 876 934
2 344 424 546
2 364 572 678
2 384 432 578
>>
>>7843168
The answer I was waiting for
>>
>>7841743
>>7841757
Hold on, did you prove that
>c-b = b-a
or
>c-b =/= b-a
?
>>
>>7843168
Wow , what the fuck have i been doing with C all these years?!
>>
>>7843606
int a, b, c, g;
for (a=2; a<1000 a+=2)
for (b=a; b<1000; b+=2)
for (c=b; c<1000; c+=2)
for (g=c; g<1000; g+=2)
if (a*a + b*b + c*c == g*g)
printf("%d %d %d %d\n", a, b, c, g);

problem?
>>
>>7843168
Improvement that reduces its execution time for big ranges.

limit = 1000
for a in xrange(2, limit, 2):
....for b in xrange(a, limit, 2):
........for c in xrange(b, limit, 2):
............for g in xrange(c, limit, 2):
................sum = a**2 + b**2 + c**2
................res = g**2
................if sum == res:
....................print(a,b,c,g)
................else:
....................if sum < res:
........................break
>>
>>7843952
Came up with a faster method to bruteforce it.
Instead of incrementing g in a loop and checking if its square is equal to the sum, just check if a^2+b^2+c^2 is a perfect square, and if it is, you have that g = sqrt(a^2+b^2+c^2) and the solution {a,b,c,g}

Python code here
http://pastebin.com/XN6Hfmme
>>
>>7843985
All the solutions until 1000, if someone's interested. With this code it took like a minute.

http://pastebin.com/GQjBg6nm
>>
>>7841389
mod 4 you fucking faggot
>>
holy FUCK what is it about python and tryhards
>>
Look it up on Wikipedia. There is a simple parameterization that generates all solutions.
>>
>>7844034
Python and brute force algorithms that add nothing to the discussion go hand in hand on /sci/.
>>
>>7843574

I see my typo between the two posts now, and thanks for your interest. I will take this as license to start sperging again. :^) But I need to back up in order to answer your question.

Your question is, "did you prove the equals-thing or the not-equals thing?" The answer is, I'm PRETTY SURE I proved a case of the the not-equals one, but I would need to review my notes first, it's been a while! I therefore won't claim to have proved it just yet, now that I'm backing up.

The claim to be proven or disproven is exactly this:

"For any given Euler Brick, the difference between its long and middle edge is never equal to that brick's difference between its middle and short edge." An equivalent, much better phrasing is "There does not exist an Euler Brick whose edge lengths are in arithmetic progression.", as I already wrote. Symbolically, the claim could be written something like: ForAll x in EB, " c - b =/= b - a ".

(Why ask this? What do you care?) What the above lemmas and this conjecture are about, is knowing just as much as one can know about the properties of Euler Bricks, /because this information relates to the open question of whether a perfect cuboid exists/. Since all perfect cuboids are by definition Euler Bricks, restrictions on what an Euler Brick may look like are restrictions on what perfect cuboids may look like. Restrictions are information.

But before I go further, I want to explain what my motivation was to investigate this particular claim.
>>
>>7844034
Python is a really easy language to teach yourself

it creates newbies with _just_ enough skill to bash out problems and the motivation to do it
>>
File: diagram.png (44 KB, 915x736) Image search: [iqdb] [SauceNao] [Google]
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>>7844276

When I started reading about Euler bricks (or EBs), I inspected a few by hand (the important c>d or d>c peculiarity), and did the above lemmas for myself. Eventually, since a brick (or cuboid) is a prosaic geometric object, it occurred to me that a brick could be thought of as resting in R^3, with one vertex at the origin.

What would it mean to think of an Euler Brick in R^3, in this way? Imagine the three-dimensional lattice points of R^3, things like (1,5,7), (8,8,8), (240,252,275), (275,240,-252), etc. In a pretty obvious sense, these lattice points are the "candidate points" for an Euler brick - so in other words, an EB can not only be thought of as a geometric cuboid, but also as a point in R^3 whose coordinates satisfy the three (Pythogorean Theorem) Diophantine equations which define an EB.

Furthermore, since we're in Cartesian-land, and admitting of negative coordinates for discussion, it also occurred to me that there were 48 different ways to orient a given EB with one vertex at the origin: the opposite vertex would always be one of 48 points, six per octant (this takes account of six permutations, as well as the 2^3 = eight different ways of signing the coordinates for a particular octant). Inspection shows, as one would expect, that all six points in a given octant are co-planar, and the intersecting plane segments give rise to a regular octahedron with vertices on the axes, being precisely (+-) (a+b+c,0,0) etc, but this isn't the important bit.
>>
>>7844315

The important part is that this geometric wondering versus the c>d OR d>c thing gave rise to consideration of other critical lines. In the diagram, we're just staying in the strictly positive octant, and considering ways that an EBs' opposite vertex may be oriented. We also consider that the GREEN lines (or major diagonals, say) are lines where it is IMPOSSIBLE for the vertices to rest, since this implies some two coordinates are equal, which we've ruled out per the lemmas. When you look at examples of Euler bricks, one notices (just by scanning data) that their edges are never in arithmetic progression (that I've seen, anyway), which is the same thing as their representations here never being on the RED lines (or minor diagonals, say).

So in a roundabout way, and not starting in a substantive way, we do eventually come to a mathematical conjecture with a little substance. This is precisely the above conjecture.
>>
>>7844320

So I began investigating this conjecture by considering points whose coordinates have a difference of one: (28,29,30), (89,90,91). Points like those. Or, (p,p+1,p+2). What would it mean to say that such a point is an EB? It would mean exactly that the coordinates satisfy the three diophantine equations of an EB. But this turns out to be impossible, in this case.

An "Almost Isosceles Right (angled) Triangle, or AIRA", is what gets implied by the above. Two examples of these are (7,24,25) and (3,4,5) (you have to be careful not to confuse the two notations, here, and I have to be careful to point that out. Where before I was referring to points in space, the above triples are simply integer triples which satisfy the pythagorean theorem.). Specifically, the above suggests that there's two of these right next to each other, but this turns out to be impossible, since there's a one-to-one correspondence between AIRAs and Square Triangular numbers per Nyblom (and also Sierpinski). IN short, these things are really "sparse", so the above is impossible. Here's the Nyblom paper:

There are details missing (I'm just proof-sketching what I really did), but I ruled out the above points, and here's the thing about EBs: when you rule out a primitive example, you also rule out their multiples. So according to my notes, I really did take care of one CASE of this, stemming from primitive points of the form (p,p+1,p+2) and their multiples. But this is not the only case to check!

The other case, and now I remember where I left off, is when you have something like (2,5,8), a primitive case where the thing is already minimal, yet the components are in arithmetic progression. These are the other things that you would have to banish in order to actually establish the conjecture as a theorem, but this is also where I left off.
>>
>>7841273
Don't know why there's so much interest here.
>>
a b c g
8 6 24 26
8 7 56 57
8 8 4 12
8 8 14 18
8 8 31 33
8 9 12 17
8 9 72 73
8 10 40 42
8 11 16 21
8 11 92 93
8 12 9 17
8 12 24 28
8 12 51 53
8 14 8 18
8 14 64 66
8 15 0 17
8 16 2 18
8 16 11 21
8 16 16 24
8 16 38 42
8 16 79 81
8 18 96 98
8 19 4 21
8 19 40 45
8 20 25 33
8 20 56 60
8 21 48 53
8 24 6 26
8 24 12 28
8 24 27 37
8 24 36 44
8 24 78 82
8 25 20 33
8 26 32 42
8 27 24 37
8 28 49 57
8 29 88 93
8 31 8 33
8 32 1 33
8 32 26 42
8 32 64 72
8 34 56 66
8 36 3 37
8 36 24 44
8 36 63 73
8 36 81 89
8 38 16 42
8 40 10 42
8 40 19 45
8 40 44 60
8 44 5 45
8 44 40 60
8 48 21 53
8 48 66 82
8 49 28 57
8 49 64 81
8 51 12 53
8 53 76 93
8 56 7 57
8 56 20 60
8 56 34 66
8 56 70 90
8 63 36 73
8 64 14 66
8 64 32 72
8 64 49 81
8 64 67 93
8 66 48 82
8 66 72 98
8 67 64 93
8 70 56 90
8 72 9 73
8 72 66 98
8 76 53 93
8 78 24 82
8 79 16 81
8 81 36 89
8 88 29 93
8 92 11 93
8 96 18 98
9 0 0 9
9 0 12 15
9 0 40 41
9 2 6 11
9 2 42 43
9 4 48 49
9 6 2 11
9 6 18 21
9 6 58 59
9 8 12 17
9 8 72 73
9 10 90 91
9 12 0 15
9 12 8 17
9 12 20 25
9 12 36 39
9 18 6 21
9 18 18 27
9 18 38 43
>>
>>7841799
People like you make sci a shit place for any discussion. You shitpost and contribute nothing to the thread whilst people are trying to actually use sci for science and maths.
And then you people complain that there's nothing here except shitty homework threads and bait.
>>
>>7848037

I wrote all the Euler brick stuff above, and while I certainly appreciate this comment (he is right, after all), I wish to state for the record that this poster is not me. It's nice to know there's common sense elsewhere.

Furthermore, a correction: In >>7844358 , (7,24,25) is of course not an example of an "almost isosceles right triangle". A better (correct) further example of one is instead (20,21,29).