Loading ...
Sorry, an error occurred while loading the content.

Re: Pythagorean Triplets

Expand Messages
  • eharsh82
    I would like to call primes of the form a^2+b^2=prime, pythagorean primes. Can anyone proove that there are infinite pythagorean primes? How about the
    Message 1 of 19 , Feb 3, 2004
    • 0 Attachment
      I would like to call primes of the form a^2+b^2=prime, pythagorean
      primes.

      Can anyone proove that there are infinite pythagorean primes?
      How about the distribution of a prime p where a^2+b^2=p^2 ?
      What if b=a+1, then do infinite pytahgorean primes exist?

      I think something similar to how Euler prooved there are infinite
      primes may work here but I am not sure.

      Let me know!

      Thanks,
      Harsh Aggarwal




      --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:
      > How do I solve this equation. Find a and b for a given prime p.
      > What properties must p have for a solution to this equation to
      exist.
      >
      > a^2 + b^2 = 0 (mod p)
      >
      > and what if b= a+1
      >
      > Let me know!
      >
      > Thanks,
      > Harsh Aggarwal
    • Andrew Swallow
      ... Or you could just call them Gaussian primes, which is more or less what they are. You should be able to find necessary information in any introductory
      Message 2 of 19 , Feb 3, 2004
      • 0 Attachment
        --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:
        > I would like to call primes of the form a^2+b^2=prime, pythagorean
        > primes.
        >
        > Can anyone proove that there are infinite pythagorean primes?
        > How about the distribution of a prime p where a^2+b^2=p^2 ?
        > What if b=a+1, then do infinite pytahgorean primes exist?
        >
        > I think something similar to how Euler prooved there are infinite
        > primes may work here but I am not sure.

        Or you could just call them Gaussian primes, which is more or less
        what they are. You should be able to find necessary information in any
        introductory number theory book. Or on whatever websites there are,
        probably.

        As for b=a+1, well that changes it to just a single dimensional
        problem, and is probably more difficult to answer.
      • eharsh82
        If both a and b are nonzero then z=a+bi, is a Gaussian prime iff a^2+b^2 is an ordinary prime. So there is actually no name for primes of the form a^2+b^2
        Message 3 of 19 , Feb 3, 2004
        • 0 Attachment
          If both a and b are nonzero then z=a+bi, is a Gaussian prime iff
          a^2+b^2 is an ordinary prime.

          So there is actually no name for primes of the form a^2+b^2

          Harsh Aggarwal


          --- In primenumbers@yahoogroups.com, "Andrew Swallow"
          <umistphd2003@y...> wrote:
          > --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:
          > > I would like to call primes of the form a^2+b^2=prime,
          pythagorean
          > > primes.
          > >
          > > Can anyone proove that there are infinite pythagorean primes?
          > > How about the distribution of a prime p where a^2+b^2=p^2 ?
          > > What if b=a+1, then do infinite pytahgorean primes exist?
          > >
          > > I think something similar to how Euler prooved there are infinite
          > > primes may work here but I am not sure.
          >
          > Or you could just call them Gaussian primes, which is more or less
          > what they are. You should be able to find necessary information in
          any
          > introductory number theory book. Or on whatever websites there are,
          > probably.
          >
          > As for b=a+1, well that changes it to just a single dimensional
          > problem, and is probably more difficult to answer.
        • elevensmooth
          ... Every prime of the form 4n+1 is the sum of two squares. Euler first communicated the following elegant proof of this fact to Goldbach in 1749, two years
          Message 4 of 19 , Feb 3, 2004
          • 0 Attachment
            --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:
            > I would like to call primes of the form a^2+b^2=prime, pythagorean
            > primes.
            >
            > Can anyone proove that there are infinite pythagorean primes?

            "Every prime of the form 4n+1 is the sum of two squares. Euler first
            communicated the following elegant proof of this fact to Goldbach in
            1749, two years after his original proof which was rathar vague on
            this point ..."

            Fermat's Last Theorem by Edwards, Springer Verlag, 1977.
            --
            ElevenSmooth: Distributed Factoring of 2^3326400-1
            http://ElevenSmooth.com
          • eharsh82
            Elevensmooth, http://www.math.uchicago.edu/~kobotis/media/163/kim.pdf prooves there are infinite primes of the from a^2+b^2 So what about when b=a+1 Any
            Message 5 of 19 , Feb 3, 2004
            • 0 Attachment
              Elevensmooth,

              http://www.math.uchicago.edu/~kobotis/media/163/kim.pdf

              prooves there are infinite primes of the from a^2+b^2

              So what about when b=a+1

              Any thoughts on this

              Harsh Aggarwal

              --- In primenumbers@yahoogroups.com, "elevensmooth"
              <elevensmooth@y...> wrote:
              > --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:
              > > I would like to call primes of the form a^2+b^2=prime,
              pythagorean
              > > primes.
              > >
              > > Can anyone proove that there are infinite pythagorean primes?
              >
              > "Every prime of the form 4n+1 is the sum of two squares. Euler
              first
              > communicated the following elegant proof of this fact to Goldbach in
              > 1749, two years after his original proof which was rathar vague on
              > this point ..."
              >
              > Fermat's Last Theorem by Edwards, Springer Verlag, 1977.
              > --
              > ElevenSmooth: Distributed Factoring of 2^3326400-1
              > http://ElevenSmooth.com
            • Carl Devore
              ... Then it becomes a univariate quadratic polynomial with interger coefficients. No univariate polynomial of degree greater than has ever been proved to
              Message 6 of 19 , Feb 3, 2004
              • 0 Attachment
                On Wed, 4 Feb 2004, eharsh82 wrote:
                > So what about when b=a+1

                Then it becomes a univariate quadratic polynomial with interger
                coefficients. No univariate polynomial of degree greater than has ever
                been proved to produce an infinite number of primes when integers are
                substituted for the variable.
              • Carl Devore
                ... Should say ...degree greater than one has ever...
                Message 7 of 19 , Feb 3, 2004
                • 0 Attachment
                  On Tue, 3 Feb 2004, Carl Devore wrote:
                  > Then it becomes a univariate quadratic polynomial with interger
                  > coefficients. No univariate polynomial of degree greater than has ever
                  > been proved to produce an infinite number of primes when integers are
                  > substituted for the variable.

                  Should say "...degree greater than one has ever..."
                • eharsh82
                  I am not sure if this series has finite number of primes or not. I think it has infinite primes. I have found several primes in the 10000 digit category. I
                  Message 8 of 19 , Feb 3, 2004
                  • 0 Attachment
                    I am not sure if this series has finite number of primes or not.
                    I think it has infinite primes.

                    I have found several primes in the 10000 digit category.
                    I used PFGW on (10^10000+$a)^2+ (10^10000+$a+1)^2

                    I'm also working on the series:
                    (2^$a)^2+(2^$a+1)^2
                    and
                    (2^$a)^2+(2^$a-1)^2

                    It din't seem to have a problem with finding primes.

                    What do you think? let me know!

                    Harsh Aggarwal



                    -- In primenumbers@yahoogroups.com, Carl Devore <devore@m...> wrote:
                    > On Tue, 3 Feb 2004, Carl Devore wrote:
                    > > Then it becomes a univariate quadratic polynomial with interger
                    > > coefficients. No univariate polynomial of degree greater than
                    has ever
                    > > been proved to produce an infinite number of primes when integers
                    are
                    > > substituted for the variable.
                    >
                    > Should say "...degree greater than one has ever..."
                  • Andy Swallow
                    ... But that is a question you will never be able to answer, one way or another, if all you re doing is search for primes of this type using computer methods.
                    Message 9 of 19 , Feb 4, 2004
                    • 0 Attachment
                      On Wed, Feb 04, 2004 at 05:31:24AM -0000, eharsh82 wrote:
                      > I am not sure if this series has finite number of primes or not.
                      > I think it has infinite primes.

                      But that is a question you will never be able to answer, one way or
                      another, if all you're doing is search for primes of this type using
                      computer methods. So wouldn't it be more interesting to study the
                      abstract theory? Your original question was about Gaussian primes, or
                      primes congruent to 1 mod 4. That's all interesting and fairly basic
                      stuff. I would have thought that more informative answers would be found
                      in there.

                      Apologies if I'm talking rubbish. It just seems strange that on the one
                      hand you're interested in whether certain sets contain infinitely many
                      primes, yet on the other hand you're studying the sets using methods
                      guaranteed to not be able to answer the question, :-)

                      Anyway, that's my morning rant out of the way...

                      Andy
                    • eharsh82
                      Here is my proof for infiniteness of these primes if b=a+1 then we get 2*a^2+2*a+1 =p solving this a is an integer if there is a prime p such that 2*p-1=m^2 or
                      Message 10 of 19 , Feb 4, 2004
                      • 0 Attachment
                        Here is my proof for infiniteness of these primes

                        if b=a+1
                        then we get 2*a^2+2*a+1 =p
                        solving this
                        a is an integer if there is a prime p such that 2*p-1=m^2
                        or m^2+1/2 is a prime

                        The distribution of such primes would follow the distribution of
                        primes with the formula n^2+1

                        it is conjectured that such primes are infinite.

                        ----
                        Taken from primepages.com

                        Are there infinitely many primes of the form n2+1?
                        There are infinitely many of the forms n2+m2 and n2+m2+1. A more
                        general form of this conjecture is if a, b, c are relatively prime, a
                        is positive, a+b and c are not both even,and b2-4ac is not a perfect
                        square, then there are infinitely many primes an2+bn+c [HW79, p19].

                        ---


                        What do you all think?

                        Also the series I talked about, what do you think about's it
                        distribution.

                        a=2^n or 2^n-1
                        b=a+1 = 2^n+1 or 2^n

                        the series reduce to ((2^n-1)^2+1)/2 and ((2^n+1)^2+1)/2

                        What about the distribution of primes in such a series?

                        let me know!


                        Harsh Aggarwal



                        --- In primenumbers@yahoogroups.com, Andy Swallow <umistphd2003@y...>
                        wrote:
                        > On Wed, Feb 04, 2004 at 05:31:24AM -0000, eharsh82 wrote:
                        > > I am not sure if this series has finite number of primes or not.
                        > > I think it has infinite primes.
                        >
                        > But that is a question you will never be able to answer, one way or
                        > another, if all you're doing is search for primes of this type using
                        > computer methods. So wouldn't it be more interesting to study the
                        > abstract theory? Your original question was about Gaussian primes,
                        or
                        > primes congruent to 1 mod 4. That's all interesting and fairly basic
                        > stuff. I would have thought that more informative answers would be
                        found
                        > in there.
                        >
                        > Apologies if I'm talking rubbish. It just seems strange that on the
                        one
                        > hand you're interested in whether certain sets contain infinitely
                        many
                        > primes, yet on the other hand you're studying the sets using methods
                        > guaranteed to not be able to answer the question, :-)
                        >
                        > Anyway, that's my morning rant out of the way...
                        >
                        > Andy
                      • pop_stack
                        Hi, I m a newby so please be gentle. OK There is no integer solution for X^2 + Y^2 = Z^2 when X and Y are prime. (In other words, no two primes, squared and
                        Message 11 of 19 , Feb 9, 2004
                        • 0 Attachment
                          Hi, I'm a newby so please be gentle.
                          OK

                          There is no integer solution for
                          X^2 + Y^2 = Z^2 when X and Y are prime.

                          (In other words, no two primes, squared and sumed can equal a perfect
                          square).

                          This came out of my Pythagorean triplets program
                          that seems to show that either X AND/OR Z are always prime
                          and Y is never prime.

                          I am not sophisticated enough to know whether the above is trivial.

                          But, I'm excited to find a group for prime numbers.

                          BTW, I have Visual Basic and or Excel demonstrations of
                          the statements above. No proofs, of course.

                          Thanks for any input.

                          pop_stack

                          leppart@...
                        • Jud McCranie
                          ... This holds if x and y are both odd (not just prime) because x^2+y^2 must be congruent to 2 mod 4 and z^2 must be congruent to 0 mod 4. It holds when x=2
                          Message 12 of 19 , Feb 9, 2004
                          • 0 Attachment
                            At 11:30 PM 2/9/2004, pop_stack wrote:
                            >Hi, I'm a newby so please be gentle.
                            >OK
                            >
                            >There is no integer solution for
                            >X^2 + Y^2 = Z^2 when X and Y are prime.

                            This holds if x and y are both odd (not just prime) because x^2+y^2 must be
                            congruent to 2 mod 4 and z^2 must be congruent to 0 mod 4. It holds when
                            x=2 (more generally when x == 2 mod 4) and y is odd for similar reasons.
                          • eharsh82
                            I realized that the series (2^n+1)^2 + (2^n)^2 and (2^n-1)^2 + (2^n) ^2 are nothing but Aurifeuillian Factors. They have some special properties that I have
                            Message 13 of 19 , Feb 25, 2004
                            • 0 Attachment
                              I realized that the series (2^n+1)^2 + (2^n)^2 and (2^n-1)^2 + (2^n)
                              ^2 are nothing but Aurifeuillian Factors.

                              They have some special properties that I have discovered.

                              1) Basically these numbers are 2^p+2^((p+1)/2)+1 and 2^p-2^((p+1)/2)+1

                              2^(2p)+1=(2^p+2^((p+1)/2)+1)*(2^p-2^((p+1)/2)+1)
                              2^(2p)+1= M2p * L2p (Notation used in literature)

                              2) p must be prime so that either L or M can be a base 2-PRP.

                              3) Either L or M is divisible by 5. When p%8=1 or 5 M is divisible by
                              5 and when p%8=3 or 7 L is divisible by 5. So only one candidate
                              remains for each prime.

                              4) So the factors of these numbers are of the form 4*p*k+1.

                              5) I think their distribution is really similar to mersenne primes,
                              as most of their properties. I have searched these numbers up to
                              p=35000 and am continuing to search higher. I have found them to
                              produce an equal number of primes as mersenne numbers. I think these
                              primes are the Gaussian equivalents of mersenne primes.

                              6) I think a top 20 list of these numbers can be started on the
                              primepages.org web page since these numbers are well known and have
                              been discussed in a lot of papers.

                              7) I am not sure if DWT can be used productively, with this series.
                              But if anyone knows how it can be used productively, please let me
                              know.

                              In order to speed up the search to higher n's, I am looking for a
                              sieve/ Trial factorer. Could someone with the required skill please
                              write me a program to sieve? I did try myself to write one but it is
                              not very fast. I am currently using that and sieving all numbers up
                              to 25G before moving to PRPing. (takes about 30 sec to take a
                              candidate to 25 G)

                              Let me know, if any one can help.

                              -- Harsh Aggarwal


                              Here are the primes I have found so far.

                              2^1+2^((1+1)/2)+1
                              2^1-2^((1+1)/2)+1
                              - Complete Set -
                              2^3+2^((3+1)/2)+1
                              2^3-2^((3+1)/2)+1
                              - Complete Set -
                              2^5+2^((5+1)/2)+1
                              2^7-2^((7+1)/2)+1
                              2^11+2^((11+1)/2)+1
                              2^19+2^((19+1)/2)+1
                              2^29+2^((29+1)/2)+1
                              2^47-2^((47+1)/2)+1
                              2^73-2^((73+1)/2)+1
                              2^79-2^((79+1)/2)+1
                              2^113-2^((113+1)/2)+1
                              2^151-2^((151+1)/2)+1
                              2^157+2^((157+1)/2)+1
                              2^163+2^((163+1)/2)+1
                              2^167-2^((167+1)/2)+1
                              2^239-2^((239+1)/2)+1
                              2^241-2^((241+1)/2)+1
                              2^283+2^((283+1)/2)+1
                              2^353-2^((353+1)/2)+1
                              2^367-2^((367+1)/2)+1
                              2^379+2^((379+1)/2)+1
                              2^457-2^((457+1)/2)+1
                              2^997+2^((997+1)/2)+1
                              2^1367-2^((1367+1)/2)+1
                              2^3041-2^((3041+1)/2)+1
                              2^10141+2^((10141+1)/2)+1
                              2^14699+2^((14699+1)/2)+1
                              2^27529-2^((27529+1)/2)+1

                              ----------------------------------------------------------------------




                              --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:
                              > Here is my proof for infiniteness of these primes
                              >
                              > if b=a+1
                              > then we get 2*a^2+2*a+1 =p
                              > solving this
                              > a is an integer if there is a prime p such that 2*p-1=m^2
                              > or m^2+1/2 is a prime
                              >
                              > The distribution of such primes would follow the distribution of
                              > primes with the formula n^2+1
                              >
                              > it is conjectured that such primes are infinite.
                              >
                              > ----
                              > Taken from primepages.com
                              >
                              > Are there infinitely many primes of the form n2+1?
                              > There are infinitely many of the forms n2+m2 and n2+m2+1. A more
                              > general form of this conjecture is if a, b, c are relatively prime,
                              a
                              > is positive, a+b and c are not both even,and b2-4ac is not a
                              perfect
                              > square, then there are infinitely many primes an2+bn+c [HW79, p19].
                              >
                              > ---
                              >
                              >
                              > What do you all think?
                              >
                              > Also the series I talked about, what do you think about's it
                              > distribution.
                              >
                              > a=2^n or 2^n-1
                              > b=a+1 = 2^n+1 or 2^n
                              >
                              > the series reduce to ((2^n-1)^2+1)/2 and ((2^n+1)^2+1)/2
                              >
                              > What about the distribution of primes in such a series?
                              >
                              > let me know!
                              >
                              >
                              > Harsh Aggarwal
                              >
                              >
                              >
                              > --- In primenumbers@yahoogroups.com, Andy Swallow
                              <umistphd2003@y...>
                              > wrote:
                              > > On Wed, Feb 04, 2004 at 05:31:24AM -0000, eharsh82 wrote:
                              > > > I am not sure if this series has finite number of primes or not.
                              > > > I think it has infinite primes.
                              > >
                              > > But that is a question you will never be able to answer, one way
                              or
                              > > another, if all you're doing is search for primes of this type
                              using
                              > > computer methods. So wouldn't it be more interesting to study the
                              > > abstract theory? Your original question was about Gaussian
                              primes,
                              > or
                              > > primes congruent to 1 mod 4. That's all interesting and fairly
                              basic
                              > > stuff. I would have thought that more informative answers would
                              be
                              > found
                              > > in there.
                              > >
                              > > Apologies if I'm talking rubbish. It just seems strange that on
                              the
                              > one
                              > > hand you're interested in whether certain sets contain infinitely
                              > many
                              > > primes, yet on the other hand you're studying the sets using
                              methods
                              > > guaranteed to not be able to answer the question, :-)
                              > >
                              > > Anyway, that's my morning rant out of the way...
                              > >
                              > > Andy
                            • mikeoakes2@aol.com
                              ... You are confirming known results. I introduced these Gaussian Mersennes in a post to the Mersenne mailing list in 2000:
                              Message 14 of 19 , Feb 26, 2004
                              • 0 Attachment
                                In a message dated 26/02/04 01:58:11 GMT Standard Time, harsh@... writes:


                                > I realized that the series (2^n+1)^2 + (2^n)^2 and (2^n-1)^2 + (2^n)
                                > ^2 are nothing but Aurifeuillian Factors.
                                >
                                > They have some special properties that I have discovered.
                                >
                                > 1) Basically these numbers are 2^p+2^((p+1)/2)+1 and 2^p-2^((p+1)/2)+1
                                >
                                > 2^(2p)+1=(2^p+2^((p+1)/2)+1)*(2^p-2^((p+1)/2)+1)
                                > 2^(2p)+1= M2p * L2p (Notation used in literature)
                                >
                                > 2) p must be prime so that either L or M can be a base 2-PRP.
                                >
                                > 3) Either L or M is divisible by 5. When p%8=1 or 5 M is divisible by
                                > 5 and when p%8=3 or 7 L is divisible by 5. So only one candidate
                                > remains for each prime.
                                >
                                > 4) So the factors of these numbers are of the form 4*p*k+1.
                                >
                                > 5) I think their distribution is really similar to mersenne primes,
                                > as most of their properties. I have searched these numbers up to
                                > p=35000 and am continuing to search higher. I have found them to
                                > produce an equal number of primes as mersenne numbers. I think these
                                > primes are the Gaussian equivalents of mersenne primes.
                                >
                                > 6) I think a top 20 list of these numbers can be started on the
                                > primepages.org web page since these numbers are well known and have
                                > been discussed in a lot of papers.
                                >
                                > 7) I am not sure if DWT can be used productively, with this series.
                                > But if anyone knows how it can be used productively, please let me
                                > know.
                                >
                                > In order to speed up the search to higher n's, I am looking for a
                                > sieve/ Trial factorer. Could someone with the required skill please
                                > write me a program to sieve? I did try myself to write one but it is
                                > not very fast. I am currently using that and sieving all numbers up
                                > to 25G before moving to PRPing. (takes about 30 sec to take a
                                > candidate to 25 G)
                                >
                                > Let me know, if any one can help.
                                >

                                You are confirming known results.

                                I introduced these "Gaussian Mersennes" in a post to the Mersenne mailing
                                list in 2000:
                                http://www.mail-archive.com/mersenne@.../msg05162.html

                                Chris Caldwell already has a "Top-20" page about these numbers:
                                http://primes.utm.edu/top20/page.php?id=41

                                You might also like to look at my post "Gaussian analogues of the Cullen and
                                Woodall primes" of Dec 2000:
                                http://listserv.nodak.edu/scripts/wa.exe?A2=ind0012&L=nmbrthry&P=R529&D=0&H=0&
                                O=T&T=1

                                -Mike Oakes


                                [Non-text portions of this message have been removed]
                              • Mark Rodenkirch
                                ... http://listserv.nodak.edu/scripts/wa.exe?A2=ind0012&L=nmbrthry&P=R529&D=0&H=0&O=T&T=1 I looked at your link and it is quite interesting. I have a couple
                                Message 15 of 19 , Feb 26, 2004
                                • 0 Attachment
                                  >
                                  http://listserv.nodak.edu/scripts/wa.exe?A2=ind0012&L=nmbrthry&P=R529&D=0&H=0&O=T&T=1

                                  I looked at your link and it is quite interesting. I have a couple of
                                  comments though. You mention that G0(n) = n*(1+i)^n + 1 and is
                                  related to Cullens but later state that G0(n) = n*2^(n/2) + 1 and then
                                  go on to show primes of that form. Am I missing something? n*(1+i)^n
                                  + 1 =/= n*2^(n/2) + 1. The same could be said of G2(n) and Woodalls.

                                  You also have Ne(n) as n^2*2^n + 1, which pre-dates the Hyper-Cullen
                                  search of Steven Harvey. He has noted your finds as he searches up to
                                  200000.

                                  --Mark
                                • mikeoakes2@aol.com
                                  In a message dated 26/02/04 14:20:52 GMT Standard Time, mgrogue@wi.rr.com ... It is if n = 0 mod 8, which was (one of) the values I was talking about.
                                  Message 16 of 19 , Feb 26, 2004
                                  • 0 Attachment
                                    In a message dated 26/02/04 14:20:52 GMT Standard Time, mgrogue@...
                                    writes:


                                    > You mention that G0(n) = n*(1+i)^n + 1 and is
                                    > related to Cullens but later state that G0(n) = n*2^(n/2) + 1 and then
                                    > go on to show primes of that form. Am I missing something? n*(1+i)^n
                                    > + 1 =/= n*2^(n/2) + 1.
                                    >
                                    It is if n = 0 mod 8, which was (one of) the values I was talking about.
                                    Remember: (1+i)^2 = 2*i.

                                    -Mike Oakes


                                    [Non-text portions of this message have been removed]
                                  Your message has been successfully submitted and would be delivered to recipients shortly.