I thought you needed some kind of reference from some upstanding member of the
mathematical establishment, for that one....


--- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>
>
>
> --- In primenumbers@yahoogroups.com,
> "ianredwood" <ianredwood@> asked:
>
> > Can anyone point me towards any worthwhile preprint
> > servers for papers in number theory?
>
> http://arxiv.org/list/math.NT/recent
>
> David
>

Sorry Paul,

I didn't mean to be mean to you. it's that David
character...

why do I feel like I'm wasting my time with him?
a leopard never changes its spots; when I open up
the home page of this group is clearly says...
moderated: NO!, but he can't seem to follow this
cardinal rule as he walks into nearly every dis-
cussion; he just doesn't like me, because I can't
and won't tolerate him! 'nuff said; it's Christ's
day, and I don't want to ruin it for anyone.

Bill

--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...>
wrote:
>
> ..
> > >
> >
> > 99999999 is neither prime nor not prime. What is such a number called?
> >
>
> your program works now! it is composite, Maybe I did not see the word
"composite" change after testing another composite number. It was so fast!
>
> You should have a look at factoris:
> http://wims.unice.fr/wims/en_tool~algebra~factor.html
>
> Paul
>

--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...>
wrote:
>
> ..
> > >
> >
> > 99999999 is neither prime nor not prime. What is such a number called?
> >
>
> your program works now! it is composite, Maybe I did not see the word
"composite" change after testing another composite number. It was so fast!

you can review a description of the method by typing in google...
t-sequence apparatus; the co-authors tried to patent the method,
but have an incorrect interpretation of how to prove primality, etc.
anyone can view my Adobe Flex 4 T-Sequence app. at my website...
www.oddperfectnumbers.com; click on the T-Sequence vs. ROH heading.

it's faster than Fast Eddie... at Jimmy Johns sandwich shop!
Bill

>
> You should have a look at factoris:
> http://wims.unice.fr/wims/en_tool~algebra~factor.html
>
> Paul
>

--- In primenumbers@yahoogroups.com,
Bill Bouris <leavemsg1@...> wrote:

> someone had to believe in the method

Non certum est, quia impossibile.

David

--- In primenumbers@yahoogroups.com,
"djbroadhurst" <d.broadhurst@...> wrote:

> Non certum est, quia impossibile.

Explanation: Bill's latest wriggle is this:
> A prime has to pass 2 full-period tests with 2 different seeds
> and then it can be declared prime
Yet we know composite numbers that pass *all* "T-sequence" tests.

Bill intends to
> program it in Java to allow for large integers
where large, for him, means more than 8 digits.
Then he will feel the inevitable Pinch:
http://www.chalcedon.demon.co.uk/rgep/p20.pdf

David

David,

if you are claiming to be so understandably particular,
then give me an absolute quadratic pseudo-prime which's
< 8 digits, and I will see if my program detects it as
composite, or if it slips through as you say it will...

Bill

--- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...> wrote:
>
> David,
>
> if you are claiming to be so understandably particular,
> then give me an absolute quadratic pseudo-prime which's
No idea what a "absolute quadratic pseudo-prime" is
> < 8 digits, and I will see if my program detects it as
> composite, or if it slips through as you say it will...
how about these 80189 & 84419
They seem to slip through quite nicely
up to 1000000 the discrepancy is
x        pi(x)    Bill
1000     168      168
10000    1229     1229
100000   9592     9594
1000000  78498    78503

Cheers
Ken

--- In primenumbers@yahoogroups.com,
"kraDen" <kradenken@...> wrote:

> up to 1000000 the discrepancy is
> x        pi(x)    Bill
> 1000     168      168
> 10000    1229     1229
> 100000   9592     9594
> 1000000  78498    78503

I'm waiting for a 14-digit version to give Bill the absolute Pinch.

David

--- On Tue, 10/19/10, leavemsg1 <leavemsg1@...> wrote:
> David,
>
> if you are claiming to be so understandably particular,
> then give me an absolute quadratic pseudo-prime which's
> < 8 digits, and I will see if my program detects it as
> composite, or if it slips through as you say it will...

No, that's not how maths works. You claimed to have a rule that does not depend
on the size of the number, and therefore any sized counter-example is a disproof
of your claim.

And with very few exceptions, rules that don't apply to almost all of the
possible set of inputs (i.e. only apply to an insignificant proportion) are
mathematically uninteresting.

Phil

dear kraDen,

if I could just get my act together, then my program might be
successful. Here's the final corrections for T-Sequence; if it
works, then it works... if it doesn't, then it will always fail.
the array size for A() might have to increase for larger N's.
this is the absolute final last ditch effort; note: Line 146 steps
by ones.
thanks to all that have tested it! still hoping it'll work...
I don't have a big integer package, or the internet at home even.

Bill

100 CLS : C = 1
102 DIM A(1001), T(3)
104 FOR N = 3 TO 7919 STEP 2
106   IF (N <> 3 AND N MOD 3 <> 0) AND (N <> 5 AND N MOD 5 <> 0) THEN
108     A(0) = N: A(1) = (N + 1) / 2: A(2) = A(1) - 1
110     FOR I = 3 TO 1001 STEP 2
112       IF A(I - 1) <= 2 THEN
114         IF A(I - 1) = 2 THEN
116           A(I) = 1: A(I + 1) = 0
118           M = I + 1: I = 1001
120         ELSE
122           A(I) = 0: M = I: I = 1001
124         END IF
126       ELSE
128         IF A(I - 1) MOD 2 = 1 THEN
130           A(I) = A(I - 2) / 2
132         ELSE
134           A(I) = (A(I - 2) + 1) / 2
136         END IF
138         A(I + 1) = A(I) - 1
140       END IF
142     NEXT I
144     P = 0
146     FOR L = 3 TO N - 2

I know that I told you to step by two's in this loop; ignore that

148       T(0) = 2: T(1) = L: T(2) = (T(1) ^ 2 - 2) MOD N
150       IF T(2) <= 1 THEN T(2) = T(2) + N
152       K = 0: FLAG = 0: Z1 = 0: Z2 = 0
154       FOR J = M - 3 TO 0 STEP -1
156         IF A(J) MOD 2 = 1 THEN
158           IF A(J) = A(J + 1) + A(J + 2) THEN K = 0 ELSE K = 1
160           T(3) = (T(2 - K) * T(1 - K) - L) MOD N
162         ELSE
164           T(3) = (T(1) ^ 2 - 2) MOD N
166         END IF
168         IF T(3) <= 1 THEN T(3) = T(3) + N
170         IF T(3) = 2 AND J <> 1 THEN
172           J = 0: FLAG = 1: REM period must be full for valid test
174         ELSE
176           IF J = 2 THEN
178             Z1 = (T(3) ^ 2 - 2) MOD N
180             IF Z1 <= 1 THEN Z1 = Z1 + N
182           END IF
184           IF J = 1 THEN
186             Z2 = (T(3) ^ 2 - 2) MOD N
188             IF Z2 <= 1 THEN Z2 = Z2 + N
190           END IF
192           T(0) = T(1): T(1) = T(2): T(2) = T(3)
194         END IF
196       NEXT J
198       IF FLAG <> 1 THEN
200         Q = (L ^ 2 - 2) MOD N
202         IF Q <= 1 THEN Q = Q + N
204         IF T(3) = L THEN
206           IF (Z1 = 2 AND Z2 = Q) OR (Z1 = Q AND Z2 = 2) THEN
208             FLAG = 0: P = P + 1
210           ELSE
212             PRINT N; "is composite!": L = N - 2
214           END IF
216         ELSE
218           PRINT N; "is composite!": L = N - 2
220         END IF
222       END IF
224       IF P = 2 THEN PRINT N; "is prime!,"; P: L = N - 2: C = C + 1
225       REM SLEEP 3
226     NEXT L
230   ELSE
232     IF N = 3 OR N = 5 THEN
234       PRINT N; "is prime!": C = C + 1
236     ELSE
238       PRINT N; "is composite!"
240     END IF
242   END IF
244 NEXT N
246 PRINT C
248 END

cheers, as you'd say!

David, kraDen,
I'm elaborating on the post, previous to this one:
the 2 problems that plagued the programs before this one were...
L ranges from 3 to (something larger than 25) up to N -2, but it
will never reach such a large value,
and the size of array A will grow slightly as N grows in size.
those two considerations should be the final corrections.
there is nothing else to correct; there wouldn't be any other
version than the one just posted...
Bill

--- In primenumbers@yahoogroups.com,
Bill Bouris <leavemsg1@...> wrote:

> if I could just get my act together, then my program might be
> successful

Ken has already told you that there are 5 pseudoprimes
less than 10^6. Here I write your algorithm in Pari-GP
and generate these 5, with their factorizations and
false witnesses, by exhaustion:

  {bin(N)=local(A=[N,(N+1)/2,(N-1)/2],b,c,t=1,i=1);
  while(t,i+=2;b=A[i]%2;if(A[i]<3,t=0;A=concat(A,if(b,0,[1,0])),
  c=(A[i-1]+1-b)/2;A=concat(A,[c,c-1])));A;}

  {lucas(N,A,L)=local(T=Mod([2,L,L^2-2],N),T3,K,F,Z=[],t);
  forstep(j=#A-3,1,-1,if(A[j]%2,K=if(A[j]==A[j+1]+A[j+2],0,1);
  T3=T[3-K]*T[2-K]-L,T3=T[2]^2-2);if(T3==2&&j-2,F=j;break());
  if(j>1&&j<4,Z=concat(Z,T3^2-2));T=[T[2],T[3],T3]);
  if(!F,t=T3==L&&(Z==[2,L^2-2]||Z==[L^2-2,2]));[F,t];}

  {bill(N)=local(A,L=2,P=[],t); \\ for odd N > 5
  if(gcd(N,15)==1,A=bin(N);while(#P<2,L++;t=lucas(N,A,L);
  if(!t[1],if(t[2],P=concat(P,L),P=[];break()))));P;}

  {forstep(N=7,10^6,2,P=bill(N);
  if(isprime(N)!=(#P==2),print([N,factor(N)[,1]~,P])));}

[80189, [17, 53, 89], [3, 4]]
[84419, [29, 41, 71], [3, 4]]
[721801, [601, 1201], [3, 4]]
[737471, [7, 137, 769], [3, 4]]
[873181, [661, 1321], [3, 4]]

Comments:

1) My code is modelled on Bill's and hence is very inefficient.
It is notable that Bill requires 55 Lucas tests to conclude
that 1412759 is probably prime:

  print(bill(1412759));

[39, 57]

since L = 39 and L = 57 are the smallest witnesses L > 2
such that he is prepared to accept the result.
He thus uses 110 selfridges, where BPSW would use only 3.

2) It is easy to find larger pseudoprimes.
Here are 5 greater than 2*10^9:

  fooled=[2023351681,2050864921,2099611801,2189197501,2201924341];
  for(k=1,5,N=fooled[k];print([factor(N)[,1]~,bill(N)]));

[[13, 41, 71, 127, 421], [3, 4]]
[[7, 17, 19, 521, 1741], [3, 4]]
[[7, 53, 131, 43201], [3, 4]]
[[29, 41, 829, 2221], [3, 4]]
[[33181, 66361], [3, 4]]

all with false witnesses L = 3 and L = 4.
I found these in a few seconds, by the obvious google.

4) Finally, the promised Pinch, using,
http://www.chalcedon.demon.co.uk/rgep/p20.pdf
which reveals that Bill's algorithm cannot decide on the
status of N = 443372888629441, even if we allow him
to spend 200,000 selfridges:

  fail=0;all=0;N=443372888629441;A=bin(N);
  for(L=3,10^5+2,all++;t=lucas(N,A,L);if(t[1],fail++));
  print(fail"/"all" failures to decide");

100000/100000 failures to decide

David

--- In primenumbers@yahoogroups.com,
"djbroadhurst" <d.broadhurst@...> wrote:

> 1) My code is modelled on Bill's and hence is very inefficient.
> It is notable that Bill requires 55 Lucas tests to conclude
> that 1412759 is probably prime

> 2) It is easy to find larger pseudoprimes.
> Here are 5 greater than 2*10^9:
> fooled=[2023351681,2050864921,2099611801,2189197501,2201924341]

> 4) Finally, the promised Pinch, using,
> http://www.chalcedon.demon.co.uk/rgep/p20.pdf
> which reveals that Bill's algorithm cannot decide on the
> status of N = 443372888629441, even if we allow him
> to spend 200,000 selfridges

Ah, I see that I forgot the third coffin-nail. Here it is:

3) If Bill retorts by arbitrarily requiring 3 or 4 Lucas tests,
I have more pseudoprimnes, in waiting, with which to reply.
In any case, he is doomed, by the absolute Pinch, as given in (4).

David

--- In primenumbers@yahoogroups.com,
"djbroadhurst" <d.broadhurst@...> wrote:

> I'm waiting for a 14-digit version to give Bill the absolute Pinch.

Now that Bill's "T-sequence" is encoded in Pari-GP, we may use

http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1006&L=nmbrthry&P=R2
> Concerted efforts by members of the PrimeNumbers list have
> proven that 79397009999 is the second Chebyshev number

to show just how easy it is to fool Bill, over and over again:

  {bin(N)=local(A=[N,(N+1)/2,(N-1)/2],b,c,t=1,i=1);
  while(t,i+=2;b=A[i]%2;if(A[i]<3,t=0;A=concat(A,if(b,0,[1,0])),
  c=(A[i-1]+1-b)/2;A=concat(A,[c,c-1])));A;}

  {lucas(N,A,L)=local(T=Mod([2,L,L^2-2],N),T3,K,F,Z=[],t);
  forstep(j=#A-3,1,-1,if(A[j]%2,K=if(A[j]==A[j+1]+A[j+2],0,1);
  T3=T[3-K]*T[2-K]-L,T3=T[2]^2-2);if(T3==2&&j-2,F=j;break());
  if(j>1&&j<4,Z=concat(Z,T3^2-2));T=[T[2],T[3],T3]);
  if(!F,t=T3==L&&(Z==[2,L^2-2]||Z==[L^2-2,2]));[F,t];}

  {billcount(N)=local(A=bin(N),F,P,C,t);
  for(L=3,10^5+2,t=lucas(N,A,L);if(t[1],F++,if(t[2],P++,C++)));
  print(F" failures to decide");
  print(P" declarations as PRP");
  print(C" proofs of compositeness");}

  billcount(23*29*41*43*251*269);

55886 failures to decide
44114 declarations as PRP
0 proofs of compositeness

In this 11-digit example,
N = 23*29*41*43*251*269 = 79397009999
was declared probably prime in 44114 tests and was *never* proven
to be composite. The absence of a decision in the remaining 55886
tests came from Bill's last "wriggle":
> REM period must be full for valid test
but in no way proves N to be composite, since the prime
p = 1412759 also has Bill undecided, tens of thousands of times:

  billcount(1412759);

38460 failures to decide
61540 declarations as PRP
0 proofs of compositeness

David

--- In primenumbers@yahoogroups.com,
"djbroadhurst" <d.broadhurst@...> wrote:

>  billcount(23*29*41*43*251*269);
>
> 55886 failures to decide
> 44114 declarations as PRP
> 0 proofs of compositeness

Here's a neat example, from Max Alekseyev
http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1006&L=NMBRTHRY&P=R239

  billcount(61*71*89*107*131*311*1483)

7692 failures to decide
92308 declarations as PRP
0 proofs of compositeness

Moreover, there are plenty more absolute Pinch's:
http://tech.groups.yahoo.com/group/primenumbers/message/21461
> For 144153 such solutions, see the 12 MB file
> http://physics.open.ac.uk/~dbroadhu/cert/carmrob2.txt
> obtained by Robert Gerbicz on 1 April 2009

  billcount(67*79*89*101*181*191*199*233*379*463*911*991*3457*9281);

100000 failures to decide
0 declarations as PRP
0 proofs of compositeness

My knowledge of absolute quadratic pseudoprimes led me to remark

http://tech.groups.yahoo.com/group/primenumbers/message/21891
> > Non certum est, quia impossibile.
> we know composite numbers that pass *all* "T-sequence" tests

but unfortunately Bill was still locked into his delusion
that observations on small numbers, with less than 8 digits,
tell us anything sensible about the wealth of pseudoprimes.

David (in debt, as so often, to Richard Pinch)

--- In primenumbers@yahoogroups.com,
"djbroadhurst" <d.broadhurst@...> wrote:

> In this 11-digit example,
> N = 23*29*41*43*251*269 = 79397009999
> was declared probably prime in 44114 tests and was *never* proven
> to be composite. The absence of a decision in the remaining 55886
> tests came from Bill's last "wriggle":
> > REM period must be full for valid test

I was grateful that Bill had laid out his (doomed) idea in
BASIC (Beginner's All-purpose Symbolic Instruction Code), a
language developed in the mid 60's for "a less technical user
who did not have the mathematical background of the more
traditional users and was not interested in acquiring it".
That made his idea easy to port to Pari-GP. Thanks, Bill!

On a personal note, I remark that when I had to choose a language
in which to work, also in the mid 60's, there were only two
viable alternatives for a practising physicist: Algol and
ForTran. I asked: "which is the better; which is the more used?"
The answers were: "the former; the latter". I chose the latter.
This turned out (fortuitously) to be a good decision, since
ForTran was the only language available to me later, when working
as a Post-Doc, at Stanford and CERN, in the early 70's.

I was wondering how ForTran is now regarded at CERN and hence
googled "ForTran CERN", with this result, from August 2007:

http://cerncourier.com/cws/article/cern/30873
> Fortran is the language of high-performance technical computing -
> even if this is an increasingly smaller segment of today's
> computing activities. In 1990, the former IT division leader
> Paolo Zanella wrote: "If I had to pick one thing likely to still
> be alive 30 years from now, I would choose Fortran."

That made me feel less of a dinosaur :-)

NB: This is not intended to spark a language-war;
it's just a harmless reminiscence.

David

I remember learning FORTRAN in high school, circa 1982, and
building SPICE from the FORTRAN source code, circa 1986.

At the time, 1986, I was already programming in C as a job
(college undergrad, working in a lab), and just a little bit
in awe that somebody was able to write such a complicated
program as SPICE using FORTRAN.

Of course, a few years later, I ended up writing and maintaining
an assembly language program with roughly 10,000 lines of code,
which was probably nearly as masochistic...



On 10/21/2010 9:36 AM, djbroadhurst wrote:
>
>
> --- In primenumbers@yahoogroups.com,
> "djbroadhurst"<d.broadhurst@...>  wrote:
>
>> In this 11-digit example,
>> N = 23*29*41*43*251*269 = 79397009999
>> was declared probably prime in 44114 tests and was *never* proven
>> to be composite. The absence of a decision in the remaining 55886
>> tests came from Bill's last "wriggle":
>>> REM period must be full for valid test
>
> I was grateful that Bill had laid out his (doomed) idea in
> BASIC (Beginner's All-purpose Symbolic Instruction Code), a
> language developed in the mid 60's for "a less technical user
> who did not have the mathematical background of the more
> traditional users and was not interested in acquiring it".
> That made his idea easy to port to Pari-GP. Thanks, Bill!
>
> On a personal note, I remark that when I had to choose a language
> in which to work, also in the mid 60's, there were only two
> viable alternatives for a practising physicist: Algol and
> ForTran. I asked: "which is the better; which is the more used?"
> The answers were: "the former; the latter". I chose the latter.
> This turned out (fortuitously) to be a good decision, since
> ForTran was the only language available to me later, when working
> as a Post-Doc, at Stanford and CERN, in the early 70's.
>
> I was wondering how ForTran is now regarded at CERN and hence
> googled "ForTran CERN", with this result, from August 2007:
>
> http://cerncourier.com/cws/article/cern/30873
>> Fortran is the language of high-performance technical computing -
>> even if this is an increasingly smaller segment of today's
>> computing activities. In 1990, the former IT division leader
>> Paolo Zanella wrote: "If I had to pick one thing likely to still
>> be alive 30 years from now, I would choose Fortran."
>
> That made me feel less of a dinosaur :-)
>
> NB: This is not intended to spark a language-war;
> it's just a harmless reminiscence.
>
> David
>
>
>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
> Yahoo! Groups Links
>
>
>
>
>

On Thu, 2010-10-21 at 16:36 +0000, djbroadhurst wrote:

> I was grateful that Bill had laid out his (doomed) idea in
> BASIC (Beginner's All-purpose Symbolic Instruction Code), a
> language developed in the mid 60's for "a less technical user
> who did not have the mathematical background of the more
> traditional users and was not interested in acquiring it".
> That made his idea easy to port to Pari-GP. Thanks, Bill!
>
> On a personal note, I remark that when I had to choose a language
> in which to work, also in the mid 60's, there were only two
> viable alternatives for a practising physicist: Algol and
> ForTran. I asked: "which is the better; which is the more used?"
> The answers were: "the former; the latter". I chose the latter.
> This turned out (fortuitously) to be a good decision, since
> ForTran was the only language available to me later, when working
> as a Post-Doc, at Stanford and CERN, in the early 70's.
>
> I was wondering how ForTran is now regarded at CERN and hence
> googled "ForTran CERN", with this result, from August 2007:
>
> http://cerncourier.com/cws/article/cern/30873
> > Fortran is the language of high-performance technical computing -
> > even if this is an increasingly smaller segment of today's
> > computing activities. In 1990, the former IT division leader
> > Paolo Zanella wrote: "If I had to pick one thing likely to still
> > be alive 30 years from now, I would choose Fortran."
>
> That made me feel less of a dinosaur :-)
>
> NB: This is not intended to spark a language-war;
> it's just a harmless reminiscence.
>
> David

I know of three languages still surviving from the 1950s, and thriving,
--- FORTRAN (my preferred spelling), COBOL and LISP.  I speak the 1st
and 3rd of these and have written "scientific" code in each of them.
Another, Algol-60 is extinct in the wild but its progeny include the
most widely used languages by far, including C, C++, C#, Java, Pascal,
Perl and many, many more.

FWIW, I believe FOCAL to be a far superior language to its co-eval BASIC
and that it's a great shame that BASIC was developed rather than FOCAL.

Algol68 was a truly beautiful elegant language.  It would be by far my
preferred language if it were still properly supported and if the I/O
model was brought out of the 1960s.

Ah, them were the days.  I'm getting quite soggy with nostalgia.


Paul

ObLanguageJoke: object-orientated COBOL is called "ADD ONE TO COBOL
GIVING COBOL"

--- In primenumbers@yahoogroups.com,
Phil Carmody <thefatphil@...> wrote:

> --- On Tue, 10/19/10, leavemsg1 <leavemsg1@...> wrote:
> > give me an absolute quadratic pseudo-prime which's
> > < 8 digits, and I will see if my program detects it as
> > composite
>
> No, that's not how maths works. You claimed to have a rule that
> does not depend on the size of the number, and therefore any sized
> counter-example is a disproof of your claim.

The following 130-digit counter-example is an
"absolute Pinch" constructed by Robert Gerbicz.

First we take a product of 46 primes, to construct a composite
number N with more than 70 trillion divisors. Then we grant
Bill 200,000 selfridges, to try to prove that N is composite.
His "T-sequence" tests, using the Lucas sequences V(L,1,n),
for L > 2, are powerless to do so:

  {pls = [23, 37, 41, 43, 53, 61, 71, 79, 89, 103, 109, 127, 131,
  181, 191, 199, 233, 239, 307, 373, 379, 419, 433, 449, 463, 521,
  701, 911, 929, 991, 1217, 1429, 1871, 2729, 3191, 4159, 4523,
  5279, 5851, 9281, 11969, 15313, 15809, 23561, 23869, 244529];
  N = prod(k=1,46,pls[k]);
  billcount(N);
  print(round(gettime/10^3)" seconds");}

100000 failures to decide
0 declarations as PRP
0 proofs of compositeness
389 seconds

Explanation:
N - 1 = 2^8 * 3^3 * 5^2 * 7 * 11 * 13 * 17 * 19 * 29 * 31 *
602647 * 71094574063 * 153412364640881519 * 17004228016300515863 *
1534710664740928440963893 * 613926412587513188575754592697115186059
is divisible by p^2 - 1, for every prime p that divides N.

David

In the last posting on this subject I showed how generation of primes can be
done using the above formula. The generation of primes was successful up until
the Mersenne pirme 2^86343 - 1. For the Mersenne after this it became difficult,
and hence the reason I had to modify the formula.

I am working on a document which explains all of the important steps. The
document will be worth sharing if I succeed in the generation of primes of size
100 000 digits or more, which I am presently hunting.

Below are the results showing PRPs and primes produced when applying the formula
on 2^110503 - 1.

A. 2^110503 - 2^n - 2^k - 1



2^110503-2^1-2^2222-1 is probable prime! (a = 65413) (digits:33265)

2^110503-2^1-2^2222-1 is probable prime!  (verification : a = 65413)
(digits:33265)


2^110503-2^1-2^27322-1 is probable prime! (a = 48481) (digits:33265)

2^110503-2^1-2^27322-1 is probable prime!  (verification : a = 48481)
(digits:33265)


2^110503-2^80540-2^55077-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)



2^110503-2^80541-2^90960-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)



B. 2^110503 - 2^n - 2^k - 1


2^110503+2^80541+2^66575-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

2^110503+2^80541+2^80513-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

2^110503+2^80541+2^96913-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)



C. 2^110503 - 2^n + 2^k - 1



2^110503-2^80540+2^64940-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

2^110503-2^80540+2^69229-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

2^110503-2^80542+2^78335-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

2^110503-2^80540+2^83148-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

2^110503-2^80541+2^54527-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)


2^110503-2^80541+2^88273-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

2^110503-2^80541+2^93763-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

2^110503-2^80542+2^96299-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265).



The work is getting a bit more complicated. I had to jump values of n much
greater than 1, for fast results. Thanks to the tips from David Broadhurst.



Peter.


[Non-text portions of this message have been removed]

What about the form 2^n+3^n ?
Do primes exist in this form?
There are sites about this form?
Thank you
Regards
Giovanni Di Maria

A beautifull day,

there are 22 new 30000 digit primes in the collection :-)
http://beablue.selfip.net/devalco/Collection/30000/

(needed time 30 days on 6 cores)

the chance for me to find one is approximitly 1 : 3000
i think that is not bad.

how probably is to find the next mersenne prime ?

  The collection is availible under
  http://devalco.de  16. Collection of Primes

> > the collection includes :
> > 1000 digit primes 17514 primes    chance for finding
> > 2000   "     "     5635   "          1:260
> > 3000   "     "     3984   "          1:434
> > 4000   "     "      770   "          1:620
> > 5000   "     "      385   "          1:810
> > 6000   "     "      219   "          1:1050
> > 7000   "     "      146   "
> > 8000   "     "      237   "          1:1633
> > 9000   "     "      238   "
> > 10000  "     "      113   "          1:1814
>   20000  "     "       32   "          1:2300
     30000  "     "       22   "          1:3000

  Greetings from the primes
  Bernhard

Hi Giovanni,

> What about the form 2^n+3^n ?
> Do primes exist in this form?

2^0 + 3^0 = 2 is a prime!
2^1 + 3^1 = 5 is a prime!
2^2 + 3^2 = 13 is a prime!
... proof that it works for all values of 'n' is left as an exercise for
the reader. :-)

Now, on a more serious note. The only other value of n which produces a
prime and which I know is n=4 (2^4 + 3^4 = 97). Other than that, this
form is going to be very very rare -- since if X is an odd number,
(a^X + b^X) = (a+b)(a^(X-1).b^0 - a^(X-2).b^1 + ... +/- a^0.b^(X-1)).

Thus, if 'n' has any odd factor, the result is a composite. Put slightly
differently, it'd better be a power of two if the result is to be prime.
Fermat primes would be proud of their cousins. :-)

Peter

> What about the form 2^n+3^n ?
> Do primes exist in this form?
> There are sites about this form?

They are one form of generalised fermat primes. They have algebraic factors if n
has any odd factors, and so can only be prime if n is a power of 2. (Just like
Fermat Numbers.)

Phil*

Is there any place online that collects factorizations of numbers of
the form a^(2^x)+b^(2^x) ?

Just wondering; I just spent a few CPU minutes finding the complete
factorization of the 123-digit number:

2^256 + 3^256 ==
    72222721 *
    343200070657 *
    2226198380033 *
    3376663028737 *
    1839605176202823817996787333633 *
    405549420455750246193993361998354279613273199617

and it seems like one of those things that people like to put into
online databases...

On 10/22/2010 12:45 PM, Phil Carmody wrote:
>> What about the form 2^n+3^n ?
>> Do primes exist in this form?
>> There are sites about this form?
>
> They are one form of generalised fermat primes. They have algebraic factors if
n has any odd factors, and so can only be prime if n is a power of 2. (Just like
Fermat Numbers.)
>
> Phil*
>
>
>
>
>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
> Yahoo! Groups Links
>
>
>
>
>

--- In primenumbers@yahoogroups.com,
Jack Brennen <jfb@...> wrote:

> 2^256 + 3^256 ==
>    72222721 *
>    343200070657 *
>    2226198380033 *
>    3376663028737 *
>    1839605176202823817996787333633 *
>    405549420455750246193993361998354279613273199617

That was already in
http://www.leyland.vispa.com/numth/factorization/anbn/3+2.txt
which also reveals that

2^(2^9) + 3^(2^9) ==
4043777 *
57987375533057 *
406297848379393 *
296909426637032277938768757025793 *
1440276254252769698681054259953238091664449537 *
2236335363090199099071989377913382509011409921 *
21208527892142900579393424863096904168207116229856960581407134511119839872619635\
5614721

David

Jack Brennen wrote:
> Is there any place online that collects factorizations of numbers of
> the form a^(2^x)+b^(2^x) ?

You can Google numbers.
http://www.google.com/search?q=72222721+prime gives
http://www.research.att.com/~njas/sequences/A094499 which links
http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/S0025-5718-98\
-00891-6.pdf
The paper is from 1998. See page 8.

--
Jens Kruse Andersen

Peter

2^110503-1+2^100560  is prime! (M29)
no need to stop testing at M28.

gr. Rob


   ----- Original Message -----
   From: Peter Lesala
   To: primenumbers@yahoogroups.com
   Sent: Friday, October 22, 2010 7:34 PM
   Subject: [PrimeNumbers] Modification of 2^p -1 +/- 2^n



   In the last posting on this subject I showed how generation of primes can be
done using the above formula. The generation of primes was successful up until
the Mersenne pirme 2^86343 - 1. For the Mersenne after this it became difficult,
and hence the reason I had to modify the formula.

   I am working on a document which explains all of the important steps. The
document will be worth sharing if I succeed in the generation of primes of size
100 000 digits or more, which I am presently hunting.

   Below are the results showing PRPs and primes produced when applying the
formula on 2^110503 - 1.

   A. 2^110503 - 2^n - 2^k - 1

   2^110503-2^1-2^2222-1 is probable prime! (a = 65413) (digits:33265)

   2^110503-2^1-2^2222-1 is probable prime! (verification : a = 65413)
(digits:33265)

   2^110503-2^1-2^27322-1 is probable prime! (a = 48481) (digits:33265)

   2^110503-2^1-2^27322-1 is probable prime! (verification : a = 48481)
(digits:33265)

   2^110503-2^80540-2^55077-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503-2^80541-2^90960-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   B. 2^110503 - 2^n - 2^k - 1

   2^110503+2^80541+2^66575-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503+2^80541+2^80513-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503+2^80541+2^96913-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   C. 2^110503 - 2^n + 2^k - 1

   2^110503-2^80540+2^64940-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503-2^80540+2^69229-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503-2^80542+2^78335-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503-2^80540+2^83148-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503-2^80541+2^54527-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503-2^80541+2^88273-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503-2^80541+2^93763-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265)

   2^110503-2^80542+2^96299-1 is prime! [N+1, Brillhart-Lehmer-Selfridge]
(digits:33265).

   The work is getting a bit more complicated. I had to jump values of n much
greater than 1, for fast results. Thanks to the tips from David Broadhurst.

   Peter.

   [Non-text portions of this message have been removed]





[Non-text portions of this message have been removed]

A beautifull day,

i tried to developp a prime test for all primes p = 3 mod 4
http://beablue.selfip.net/devalco/helmes_test.htm

i think that there might be counterexamples if you do not take the smallest b,
but generalize the test for all b's which are a non quadratic residue.

The good news is that the test "only" need 3 Selfridges

Nice Greetings from the primes
Bernhard

--- In primenumbers@yahoogroups.com,
"bhelmes_1" <bhelmes@...> wrote:

> let be b the smallest non quadric residium.
> Necessary: b^((p-1)/2)=p-1 mod p
> Sufficent: (b+I)^p = b^p + I^p = b - I

This looks to be hard to fool, since it is very similar to BPSW.
However it is wilful nonsense to claim that such a test
proves primality. Please replace "suffic[i]ent" by "necessary".

David