| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1324.1 | I know that's not what you meant. :-) | GUESS::DERAMO | Dan D'Eramo | Tue Nov 06 1990 10:59 | 8 |
| >> 4) what is the largest n for which a solution of x^n + y^n = z^n
>> is known ?
Positive integral solutions are known for n = 2 (for example,
x = 3, y = 4, z = 5, n = 2) but for no larger n.
Dan
:-)
|
| 1324.2 | | GUESS::DERAMO | Dan D'Eramo | Tue Nov 06 1990 11:12 | 30 |
| >> 5) since it is possible to find 6 consecutive composite
>> numbers by saying N= 6!= 6*6*5*4*3*2= 720
>> so N+2 = 2[ 6*5*4*3 +1] =722
>> N+3 = 3[ 6*5*4*2 +1] =723
>> .
>> .
>> N+6 = 6[ 5*4*3*2 +1] = 726
>> so for 10! we can come with 10 consecutive composite numbers
>> following above process. we can come with Huge caps of
>> consective composite numbers, so can I conjecture that since
>> these gaps become so immense, primes will eventually cease to
>> occure, so prime are finite .
>> can I seay for oo! we can come with oo consectivte composite numbers.
>> so prime numbers are not infinite. ?
>> whats wrong with the argument.
Actually, for N = 6!, N+2 thru N+6 make up only five, not six,
consecutive composite numbers (although N and N+1 are also
composite here, for a total of seven consecutive composites).
Although the gaps becomes arbitrarily large, they do not (as the
argument incorrectly assumes) become "everything from this point
on". As Euclid first showed, there are infinitely many primes.
In a nonstandard model of, say, Peano arithmetic, if N is
nonstandard then N!+2, N!+3, N!+4, ... N!+(any standard integer > 1),
... will all be composite, but there will still be a (nonstandard)
prime larger than all of those numbers.
Dan
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| 1324.3 | any one knows how big oo is?? | SMAUG::ABBASI | | Tue Nov 06 1990 22:47 | 12 |
| ref .1
I really , really meant to ask for largest n that x^n+y^n=z^n has
been "proved" that this formula is cant be satisfied for integers.
I read in book dated 1976 that for up to n=600, fermat last theory
has been proved.
ref .2
well dan, You affirmed something about my self I always suspected.
I cant count ! :-)
/naser
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| 1324.4 | by Sam Wagstaff Jr | HERON::BUCHANAN | combinatorial bomb squad | Wed Nov 07 1990 05:51 | 10 |
| > ref .1
> I really , really meant to ask for largest n that x^n+y^n=z^n has
> been "proved" that this formula is cant be satisfied for integers.
> I read in book dated 1976 that for up to n=600, fermat last theory
> has been proved.
According to Bob Silverman on the net recently, who knows about
this kind of thing, n is currently 150,000.
Andrew.
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| 1324.5 | for completeness | HERON::BUCHANAN | combinatorial bomb squad | Wed Nov 07 1990 05:57 | 28 |
| > 1) are there finite number of twin primes ?
No one knows
> is the largest known still this : 1,000,000,009,649
> 1,000,000,009,651
No. A note here eighteen months or so had some much bigger ones
which were (then) the record.
> 2) are there infinite number of perfect numbers? (this relates
> to mersseen (sp?) numbers , since a perfect number is of forum
> 2^p(2^(p-1) -1) , where 2^(p-1) - 1 is prime (by Euler), so
> the question can also be asked is there an infinite number
> of mersseen numbers ?
(a) It's Mersenne
(b) *Even* perfect numbers are in 1-1 correspondence with the Mersenne
primes, but it's unknown whether there is an odd perfect number.
(c) It's unknown whether there is an infinite number of Mersenne primes.
> 3) what is the largest mersseen(sp?) number known ?
A very early note in this notesfile (one of the first 10, I think)
lists the exponents of all the known Mersenne primes.
Regards,
Andrew.
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| 1324.6 | | ALLVAX::JROTH | It's a bush recording... | Wed Nov 07 1990 08:27 | 10 |
| � I really , really meant to ask for largest n that x^n+y^n=z^n has
� been "proved" that this formula is cant be satisfied for integers.
� I read in book dated 1976 that for up to n=600, fermat last theory
� has been proved.
I thought the conjecture was proved to be true for some infinite classes
of exponents (by Kummer?) but not all of them - if this is so, the correct
thing to ask is for what n do we know that it is true for all k < n.
- Jim
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| 1324.7 | | GUESS::DERAMO | Dan D'Eramo | Wed Nov 07 1990 11:08 | 12 |
| � I thought the conjecture was proved to be true for some infinite classes
� of exponents (by Kummer?) but not all of them - if this is so, the correct
� thing to ask is for what n do we know that it is true for all k < n.
Has it been shown that infinitely many primes satisfy
his criteria?
I think there are different limits n for the two cases
where n divides xyz and where n does not divide xyz (where
n is assumed to be an odd prime).
Dan
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| 1324.8 | Dir/title=fermat | CIVAGE::LYNN | Lynn Yarbrough @WNP DTN 427-5663 | Wed Nov 07 1990 15:57 | 4 |
| > 4) what is the largest n for which a solution of x^n + y^n = z^n
> is known ?
See note 445.
|