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Old 2006-02-21, 16:59   #23
mfgoode
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Quote:
Originally Posted by alpertron
The trivial cases are:

Odd numbers: 2n+1 = (n+1)^2 - n^2

Numbers multiple of 4: 4n = (n+1)^2 - (n-1)^2
To put it practically I have found from my theory (similar to yours) that if one is given an odd number to be put as the difference of two squares just divide the number by two and get the squares of the integral values one lower, the other higher and subtract.

Its easier when you actually do it. Take for instance the number 125.
Half of this is 62.5 Discard the decimal and take the lower integral value as 62.Similarly the higher value is 63 so
63^2 - 62^2 = 125

Theorem:1: The square of a number is either divisible by 4 or leaves the remainder 1 when divided by 4

Theorem 2: The square of an odd number is of the form 8q + 1

There are interesting proofs of these theorems. Try them out.
Mally
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Old 2006-02-22, 14:08   #24
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Quote:
Originally Posted by mfgoode
Theorem:1: The square of a number is either divisible by 4 or leaves the remainder 1 when divided by 4

Theorem 2: The square of an odd number is of the form 8q + 1
Starting with an even integer 2n, its square is: 4n2, a multiple of 4.

Starting with an odd integer 2n+1, its square is: (2n+1)2 = 4n2+4n+1. Since n2 has the same parity of n, this means that n2 + n is even, so 4n2 + 4n is multiple of 8. Thus the square has the form 8q+1, as expected.
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Old 2006-02-23, 09:54   #25
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Quote:
Originally Posted by Numbers
And I note that none of you were able to define N
There's a definition right in the heading of A074981:

"Conjectured list of positive numbers which are not of the form r^i-s^j, where r,s,i,j are integers with r>0, s>0, i>1, j>1"

Quote:
a conundrum that arises out of my suspicion that you would not in fact be able to define N.
I think each of Silverman, xilman, and alpertron were quite able to create a similar definition, had already done so mentally, and probably considered it fairly obvious.

Quote:
Now, let f(x) = x^a - y^b, so that the problem can be stated as: When does f(x) = N? The conjecture referred to in Alpertron's post says that f(x) never = N for N in A074981.

For obvious reasons, A074981 is a finite representation of what is presumably an infinite sequence. So there are values of N that should be in the sequence that are not at OEIS, or any other list of the sequence. Let's say M is a number that should be in N, but is not on any recorded list of N.

Joe Nobody finds that f(x) = M.
But if f(x) = M, then M is not a number that "should" be in list N. List N is the list of non-solutions to f(x) = N (integer variable), not the list of solutions.

So "M is a number that should be in N" is contradictory to "Joe Nobody finds that f(x) = M", and your conclusions after that are invalid.

Quote:
Since no one can define N,
False. See above.

Quote:
how does Joe even prove that M is in N
He doesn't. No such M exists. Perhaps Joe has made a mistake in his calculations.

Quote:
that he has found a counter-example to the conjecture?
The statement that he has found a counter-example depends upon no one's noticing that you've contradicted yourself (or, maybe, that M has contradicted itself -- or that Joe has contradicted himself -- or maybe that Joe just plain made a mistake). :-)

Quote:
You could probably get quite cute with your semantics and claim that since M should be in N, then Joe cant find that f(x) = M, but I'm sure you know what I mean.
Where does self-contradiction rank on the semantic cuteness scale? :-)

Perhaps you'd care to re-state what you mean?
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Old 2006-02-23, 17:27   #26
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Quote:
Originally Posted by ewmayer
If I read Bob's post correctly, he meant that "all problematic numbers are == 2 mod 4," not "all numbers that == 2 mod 4 are problematic."
Yes.
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Old 2006-02-23, 17:31   #27
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Quote:
Originally Posted by alpertron
See the sequence at OEIS: http://www.research.att.com/~njas/sequences/A074981 : 6, 14, 34, 42, 50, 58, 62, 66, 70, 78, 82, 86, 90, 102, 110, 114, 130, 134, 158, 178, 182, 202, 206, 210, 226, 230, 238, 246, 254, 258, 266, 274, 278, 290, 302, 306, 310, 314, 322, 326, 330, 358, 374, 378, 390, 394, 398, 402, 410, 418, 422, 426, ...

According to that page it is conjectured that none of these numbers can be a difference of two powers.
Also, the density of the above sequence is unknown.
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Old 2006-02-23, 17:39   #28
ewmayer
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Quote:
Originally Posted by Numbers
I note that none of you were able to define N
You're confusing "define" with "produce every example of," the latter being by definition impossible when one is dealing with a (presumably or provably) infinite sequence. Analogously we can perfectly well *define* what a prime number is, and can even prove wonderful results about the infinitude and density thereof, without ever being able to enumerate them all.
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Old 2006-02-24, 23:32   #29
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Quote:
Originally Posted by pacionet
I have in mind (for the future, if I have lot, lot, lot time) to think, write and run a small ( but working ) distributed computing project (maybe with BOINC platform). Can you suggest me some possible project yet-to-do ? (any matter: math, games, puzzles, etc. )
Does it sound interresting to you : ?
Quote:
Originally Posted by http://www.norvig.com/beal.html
we need a new approach. My first idea is to only do a slice of the zr values at a time. This would require switching from an approach that limits the bases and powers to one that limits the minimum and maximum sum searched for. That is, we would call something like beal2(10 ** 20, 10 ** 50) to search for all solutions with sum between 1020 and 1050. The program would build a table of all zr values in that range, and then carefully enumerate x,m,y,n to stay within that range. One could then take this program and add a protocol like SETI@home where interested people could download the code, be assigned min and max values to search on, and a large network of people could together search for solutions.
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Old 2006-02-25, 03:07   #30
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Quote:
Originally Posted by cheesehead
There's a definition right in the heading of A074981:

"Conjectured list of positive numbers which are not of the form r^i-s^j, where r,s,i,j are integers with r>0, s>0, i>1, j>1"
So there's no proof that any of those numbers is not the difference of two power? It's just that we believe that some numbers belong on the list because we have searched up to some very high lower bound?
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Old 2006-03-06, 23:20   #31
ewmayer
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By popular request, I've split the thread - the theological debate which began at this point has been moved to the Soapbox.

Please keep the posts in this thread on-topic (i.e. mathematically oriented).
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Old 2006-03-06, 23:47   #32
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Actually some of the later posts were math related also.
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Old 2006-03-16, 19:28   #33
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Quote:
Originally Posted by R.D. Silverman
An open problem is whether every integer can be written as the difference
or two powers. Clearly, numbers that are odd or 0 mod 4 always have a trivial
representation. Numbers that are 2 mod 4 are the problem.

There are no known solutions to x^a - y^b = N for a,b >= 2 and
N = 6,14,50, ..... etc.
FWIW, in case somebody starts on this, I already tested the following "ranges" (there were no solutions)

N in (6,14, ... 426)

b < 32, b prime, y < 4*10^7, any x, any a
and
b < 4000, b prime, y< 400, any x, any a
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