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FW: Re: [PrimeNumbers] 999-digit prime factoid

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  • James Merickel
    ... From: merk7777777@yahoo.com To: maximilian.hasler@gmail.com Sent: Wed Aug 10th, 2011 2:36 PM EDT Subject: Re: [PrimeNumbers] 999-digit prime factoid
    Message 1 of 6 , Aug 10, 2011
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      ----Forwarded Message----
      From: merk7777777@...
      To: maximilian.hasler@...
      Sent: Wed Aug 10th, 2011 2:36 PM EDT
      Subject: Re: [PrimeNumbers] 999-digit prime factoid

      Correction placed in final sentence here.
      Three 333-digit consecutive primes are apt to exist such that their concatenation in some order is greatest of all non-titanic concatenations (Perhaps the specifics are interesting, but I didn't pursue it). Until n=9, then, there are no 999-digit prime concatenations of n consecutive primes other than the four I thought I described in (interesting enough) detail: Five must be 143 digits and the other two are those immediately under 10^142; and the reason I bothered with posting this is that three of these prime concatenations--the largest of them, I THOUGHT, placing all five of the ones with leading digit 1 left to right from largest to smallest, have the larger of the ones with leading digit 9 in initial position and the smaller one in final position. f still unclear, let me know. FWIW, I don't see much interesting in the seven primes involved other than that the second prime exceeding 10^142 does so by 111. In fact, THE FIRST IS ACTUALLY BEFORE
      THIS IN THE NUMBER THAT LOOKED INTERESTING TO ME, so I made a misstatement in posting this (Likely I would have thought it still worth a mention, but perhaps not). I can reproduce the numbers if actually wanted.
      Jim

      On Wed Aug 10th, 2011 11:26 AM EDT Maximilian Hasler wrote:

      >Dear Jim,
      >
      >for most mathematicians it takes quite some time to translate your
      >statements to formulae which give them a meaning.
      >
      >Could you be so kind as to use conventional mathematical language and
      >explicit formulae
      >instead (or at least in addition) to the "poetical" formulation ?
      >
      >below some concrete questions.
      >
      >On Wed, Aug 10, 2011 at 3:59 AM, James Merickel <merk7777777@...> wrote:
      >> Other than for concatenations of 3 of them,
      >
      >which three ?
      >
      >> the largest prime non-titanic concatenation of consecutive primes
      >
      >can you give it more explicitely ?
      >
      >> brackets
      >
      >how can ONE thing bracket something else ?
      >AFAIK you need two things to bracket something in-between !?
      >
      >> the five smallest
      >
      >could you give them ?
      >
      >> over 10^142
      >
      >what does this mean, "over" ?
      >
      >> in decreasing left-to-right order between the two largest under (also in decreasing order).
      >
      >the 2 largest under what ? can you give the values ?
      >
      >>  The next three largest are also concatenations of 7,
      >
      >what means " concatenations of 7 " ?
      >
      >> with the only one leading with the smallest
      >
      >what do you mean ?
      >
      >> having the second smallest in the 3rd position left-to-right.
      >
      >please write the chain of inequalities, it would be so much easier to
      >visualize and understand.
      >
      >Thanks in advance,
      >
      >Maximilian
    • James Merickel
      Explicitly, using the notation of http://en.wikipedia.org/wiki/Concatenation_(mathematics) , the number I misread slightly is
      Message 2 of 6 , Aug 10, 2011
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        Explicitly, using the notation of http://en.wikipedia.org/wiki/Concatenation_(mathematics) , the number I misread slightly is
        (10^142-519)||(10^142+1563)||(10^142+1083)||(10^142+301)||(10^142+13)||(10^142+111)||(10^142-563)

        ----Forwarded Message----
        From: merk7777777@...
        To: primenumbers@yahoogroups.com
        Sent: Wed Aug 10th, 2011 2:39 PM EDT
        Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid



        ----Forwarded Message----
        From: merk7777777@...
        To: maximilian.hasler@...
        Sent: Wed Aug 10th, 2011 2:36 PM EDT
        Subject: Re: [PrimeNumbers] 999-digit prime factoid

        Correction placed in final sentence here.
        Three 333-digit consecutive primes are apt to exist such that their concatenation in some order is greatest of all non-titanic concatenations (Perhaps the specifics are interesting, but I didn't pursue it). Until n=9, then, there are no 999-digit prime concatenations of n consecutive primes other than the four I thought I described in (interesting enough) detail: Five must be 143 digits and the other two are those immediately under 10^142; and the reason I bothered with posting this is that three of these prime concatenations--the largest of them, I THOUGHT, placing all five of the ones with leading digit 1 left to right from largest to smallest, have the larger of the ones with leading digit 9 in initial position and the smaller one in final position. f still unclear, let me know. FWIW, I don't see much interesting in the seven primes involved other than that the second prime exceeding 10^142 does so by 111. In fact, THE FIRST IS ACTUALLY BEFORE
        THIS IN THE NUMBER THAT LOOKED INTERESTING TO ME, so I made a misstatement in posting this (Likely I would have thought it still worth a mention, but perhaps not). I can reproduce the numbers if actually wanted.
        Jim

        On Wed Aug 10th, 2011 11:26 AM EDT Maximilian Hasler wrote:

        >Dear Jim,
        >
        >for most mathematicians it takes quite some time to translate your
        >statements to formulae which give them a meaning.
        >
        >Could you be so kind as to use conventional mathematical language and
        >explicit formulae
        >instead (or at least in addition) to the "poetical" formulation ?
        >
        >below some concrete questions.
        >
        >On Wed, Aug 10, 2011 at 3:59 AM, James Merickel <merk7777777@...> wrote:
        >> Other than for concatenations of 3 of them,
        >
        >which three ?
        >
        >> the largest prime non-titanic concatenation of consecutive primes
        >
        >can you give it more explicitely ?
        >
        >> brackets
        >
        >how can ONE thing bracket something else ?
        >AFAIK you need two things to bracket something in-between !?
        >
        >> the five smallest
        >
        >could you give them ?
        >
        >> over 10^142
        >
        >what does this mean, "over" ?
        >
        >> in decreasing left-to-right order between the two largest under (also in decreasing order).
        >
        >the 2 largest under what ? can you give the values ?
        >
        >>  The next three largest are also concatenations of 7,
        >
        >what means " concatenations of 7 " ?
        >
        >> with the only one leading with the smallest
        >
        >what do you mean ?
        >
        >> having the second smallest in the 3rd position left-to-right.
        >
        >please write the chain of inequalities, it would be so much easier to
        >visualize and understand.
        >
        >Thanks in advance,
        >
        >Maximilian




        [Non-text portions of this message have been removed]
      • James Merickel
        Sorry, link was missing open parenthesis. http://en.wikipedia.org/wiki/Concatenation_(mathematics) Jim ... From: merk7777777@yahoo.com To:
        Message 3 of 6 , Aug 10, 2011
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          Sorry, link was missing open parenthesis. http://en.wikipedia.org/wiki/Concatenation_(mathematics)
          Jim

          ----Forwarded Message----
          From: merk7777777@...
          To: primenumbers@yahoogroups.com
          Sent: Wed Aug 10th, 2011 2:55 PM EDT
          Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid

          Explicitly, using the notation of http://en.wikipedia.org/wiki/Concatenation_(mathematics) , the number I misread slightly is
          (10^142-519)||(10^142+1563)||(10^142+1083)||(10^142+301)||(10^142+13)||(10^142+111)||(10^142-563)

          ----Forwarded Message----
          From: merk7777777@...
          To: primenumbers@yahoogroups.com
          Sent: Wed Aug 10th, 2011 2:39 PM EDT
          Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid



          ----Forwarded Message----
          From: merk7777777@...
          To: maximilian.hasler@...
          Sent: Wed Aug 10th, 2011 2:36 PM EDT
          Subject: Re: [PrimeNumbers] 999-digit prime factoid

          Correction placed in final sentence here.
          Three 333-digit consecutive primes are apt to exist such that their concatenation in some order is greatest of all non-titanic concatenations (Perhaps the specifics are interesting, but I didn't pursue it). Until n=9, then, there are no 999-digit prime concatenations of n consecutive primes other than the four I thought I described in (interesting enough) detail: Five must be 143 digits and the other two are those immediately under 10^142; and the reason I bothered with posting this is that three of these prime concatenations--the largest of them, I THOUGHT, placing all five of the ones with leading digit 1 left to right from largest to smallest, have the larger of the ones with leading digit 9 in initial position and the smaller one in final position. f still unclear, let me know. FWIW, I don't see much interesting in the seven primes involved other than that the second prime exceeding 10^142 does so by 111. In fact, THE FIRST IS ACTUALLY BEFORE
          THIS IN THE NUMBER THAT LOOKED INTERESTING TO ME, so I made a misstatement in posting this (Likely I would have thought it still worth a mention, but perhaps not). I can reproduce the numbers if actually wanted.
          Jim

          On Wed Aug 10th, 2011 11:26 AM EDT Maximilian Hasler wrote:

          >Dear Jim,
          >
          >for most mathematicians it takes quite some time to translate your
          >statements to formulae which give them a meaning.
          >
          >Could you be so kind as to use conventional mathematical language and
          >explicit formulae
          >instead (or at least in addition) to the "poetical" formulation ?
          >
          >below some concrete questions.
          >
          >On Wed, Aug 10, 2011 at 3:59 AM, James Merickel <merk7777777@...> wrote:
          >> Other than for concatenations of 3 of them,
          >
          >which three ?
          >
          >> the largest prime non-titanic concatenation of consecutive primes
          >
          >can you give it more explicitely ?
          >
          >> brackets
          >
          >how can ONE thing bracket something else ?
          >AFAIK you need two things to bracket something in-between !?
          >
          >> the five smallest
          >
          >could you give them ?
          >
          >> over 10^142
          >
          >what does this mean, "over" ?
          >
          >> in decreasing left-to-right order between the two largest under (also in decreasing order).
          >
          >the 2 largest under what ? can you give the values ?
          >
          >>  The next three largest are also concatenations of 7,
          >
          >what means " concatenations of 7 " ?
          >
          >> with the only one leading with the smallest
          >
          >what do you mean ?
          >
          >> having the second smallest in the 3rd position left-to-right.
          >
          >please write the chain of inequalities, it would be so much easier to
          >visualize and understand.
          >
          >Thanks in advance,
          >
          >Maximilian




          [Non-text portions of this message have been removed]
        • James Merickel
          Another correction: The sentence claiming there are no 999-digit primes of other lengths than discussed should say with leading digit 9 as well. Jim
          Message 4 of 6 , Aug 10, 2011
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            Another correction: The sentence claiming there are no 999-digit primes of other lengths than discussed should say 'with leading digit 9' as well.
            Jim

            On Wed Aug 10th, 2011 3:00 PM EDT James Merickel wrote:

            >Sorry, link was missing open parenthesis. http://en.wikipedia.org/wiki/Concatenation_(mathematics)
            >Jim
            >
            >----Forwarded Message----
            >From: merk7777777@...
            >To: primenumbers@yahoogroups.com
            >Sent: Wed Aug 10th, 2011 2:55 PM EDT
            >Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid
            >
            >Explicitly, using the notation of http://en.wikipedia.org/wiki/Concatenation_(mathematics) , the number I misread slightly is
            >(10^142-519)||(10^142+1563)||(10^142+1083)||(10^142+301)||(10^142+13)||(10^142+111)||(10^142-563)
            >
            >----Forwarded Message----
            >From: merk7777777@...
            >To: primenumbers@yahoogroups.com
            >Sent: Wed Aug 10th, 2011 2:39 PM EDT
            >Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid
            >
            >
            >
            >----Forwarded Message----
            >From: merk7777777@...
            >To: maximilian.hasler@...
            >Sent: Wed Aug 10th, 2011 2:36 PM EDT
            >Subject: Re: [PrimeNumbers] 999-digit prime factoid
            >
            >Correction placed in final sentence here.
            >Three 333-digit consecutive primes are apt to exist such that their concatenation in some order is greatest of all non-titanic concatenations (Perhaps the specifics are interesting, but I didn't pursue it). Until n=9, then, there are no 999-digit prime concatenations of n consecutive primes other than the four I thought I described in (interesting enough) detail: Five must be 143 digits and the other two are those immediately under 10^142; and the reason I bothered with posting this is that three of these prime concatenations--the largest of them, I THOUGHT, placing all five of the ones with leading digit 1 left to right from largest to smallest, have the larger of the ones with leading digit 9 in initial position and the smaller one in final position. f still unclear, let me know. FWIW, I don't see much interesting in the seven primes involved other than that the second prime exceeding 10^142 does so by 111. In fact, THE FIRST IS ACTUALLY BEFORE
            > THIS IN THE NUMBER THAT LOOKED INTERESTING TO ME, so I made a misstatement in posting this (Likely I would have thought it still worth a mention, but perhaps not). I can reproduce the numbers if actually wanted.
            >Jim
            >
            >On Wed Aug 10th, 2011 11:26 AM EDT Maximilian Hasler wrote:
            >
            >>Dear Jim,
            >>
            >>for most mathematicians it takes quite some time to translate your
            >>statements to formulae which give them a meaning.
            >>
            >>Could you be so kind as to use conventional mathematical language and
            >>explicit formulae
            >>instead (or at least in addition) to the "poetical" formulation ?
            >>
            >>below some concrete questions.
            >>
            >>On Wed, Aug 10, 2011 at 3:59 AM, James Merickel <merk7777777@...> wrote:
            >>> Other than for concatenations of 3 of them,
            >>
            >>which three ?
            >>
            >>> the largest prime non-titanic concatenation of consecutive primes
            >>
            >>can you give it more explicitely ?
            >>
            >>> brackets
            >>
            >>how can ONE thing bracket something else ?
            >>AFAIK you need two things to bracket something in-between !?
            >>
            >>> the five smallest
            >>
            >>could you give them ?
            >>
            >>> over 10^142
            >>
            >>what does this mean, "over" ?
            >>
            >>> in decreasing left-to-right order between the two largest under (also in decreasing order).
            >>
            >>the 2 largest under what ? can you give the values ?
            >>
            >>>  The next three largest are also concatenations of 7,
            >>
            >>what means " concatenations of 7 " ?
            >>
            >>> with the only one leading with the smallest
            >>
            >>what do you mean ?
            >>
            >>> having the second smallest in the 3rd position left-to-right.
            >>
            >>please write the chain of inequalities, it would be so much easier to
            >>visualize and understand.
            >>
            >>Thanks in advance,
            >>
            >>Maximilian
            >
            >
            >
            >
            >[Non-text portions of this message have been removed]
            >
          • James Merickel
            Message 5 of 6 , Aug 10, 2011
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              On Wed Aug 10th, 2011 4:22 PM EDT James Merickel wrote:

              >http://en.wikipedia.org/wiki/Concatenation_(mathematics) [cellphone chooses word I intend typing, and assumed parenthesis was typo]
              >
              >On Wed Aug 10th, 2011 3:07 PM EDT James Merickel wrote:
              >
              >>Another correction: The sentence claiming there are no 999-digit primes of other lengths than discussed should say 'with leading digit 9' as well.
              >>Jim
              >>
              >>On Wed Aug 10th, 2011 3:00 PM EDT James Merickel wrote:
              >>
              >>>Sorry, link was missing open parenthesis. http://en.wikipedia.org/wiki/Concatenation_(mathematics)
              >>>Jim
              >>>
              >>>----Forwarded Message----
              >>>From: merk7777777@...
              >>>To: primenumbers@yahoogroups.com
              >>>Sent: Wed Aug 10th, 2011 2:55 PM EDT
              >>>Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid
              >>>
              >>>Explicitly, using the notation of http://en.wikipedia.org/wiki/Concatenation_(mathematics) , the number I misread slightly is
              >>>(10^142-519)||(10^142+1563)||(10^142+1083)||(10^142+301)||(10^142+13)||(10^142+111)||(10^142-563)
              >>>
              >>>----Forwarded Message----
              >>>From: merk7777777@...
              >>>To: primenumbers@yahoogroups.com
              >>>Sent: Wed Aug 10th, 2011 2:39 PM EDT
              >>>Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid
              >>>
              >>>
              >>>
              >>>----Forwarded Message----
              >>>From: merk7777777@...
              >>>To: maximilian.hasler@...
              >>>Sent: Wed Aug 10th, 2011 2:36 PM EDT
              >>>Subject: Re: [PrimeNumbers] 999-digit prime factoid
              >>>
              >>>Correction placed in final sentence here.
              >>>Three 333-digit consecutive primes are apt to exist such that their concatenation in some order is greatest of all non-titanic concatenations (Perhaps the specifics are interesting, but I didn't pursue it). Until n=9, then, there are no 999-digit prime concatenations of n consecutive primes other than the four I thought I described in (interesting enough) detail: Five must be 143 digits and the other two are those immediately under 10^142; and the reason I bothered with posting this is that three of these prime concatenations--the largest of them, I THOUGHT, placing all five of the ones with leading digit 1 left to right from largest to smallest, have the larger of the ones with leading digit 9 in initial position and the smaller one in final position. f still unclear, let me know. FWIW, I don't see much interesting in the seven primes involved other than that the second prime exceeding 10^142 does so by 111. In fact, THE FIRST IS ACTUALLY BEFORE
              >>> THIS IN THE NUMBER THAT LOOKED INTERESTING TO ME, so I made a misstatement in posting this (Likely I would have thought it still worth a mention, but perhaps not). I can reproduce the numbers if actually wanted.
              >>>Jim
              >>>
              >>>On Wed Aug 10th, 2011 11:26 AM EDT Maximilian Hasler wrote:
              >>>
              >>>>Dear Jim,
              >>>>
              >>>>for most mathematicians it takes quite some time to translate your
              >>>>statements to formulae which give them a meaning.
              >>>>
              >>>>Could you be so kind as to use conventional mathematical language and
              >>>>explicit formulae
              >>>>instead (or at least in addition) to the "poetical" formulation ?
              >>>>
              >>>>below some concrete questions.
              >>>>
              >>>>On Wed, Aug 10, 2011 at 3:59 AM, James Merickel <merk7777777@...> wrote:
              >>>>> Other than for concatenations of 3 of them,
              >>>>
              >>>>which three ?
              >>>>
              >>>>> the largest prime non-titanic concatenation of consecutive primes
              >>>>
              >>>>can you give it more explicitely ?
              >>>>
              >>>>> brackets
              >>>>
              >>>>how can ONE thing bracket something else ?
              >>>>AFAIK you need two things to bracket something in-between !?
              >>>>
              >>>>> the five smallest
              >>>>
              >>>>could you give them ?
              >>>>
              >>>>> over 10^142
              >>>>
              >>>>what does this mean, "over" ?
              >>>>
              >>>>> in decreasing left-to-right order between the two largest under (also in decreasing order).
              >>>>
              >>>>the 2 largest under what ? can you give the values ?
              >>>>
              >>>>>  The next three largest are also concatenations of 7,
              >>>>
              >>>>what means " concatenations of 7 " ?
              >>>>
              >>>>> with the only one leading with the smallest
              >>>>
              >>>>what do you mean ?
              >>>>
              >>>>> having the second smallest in the 3rd position left-to-right.
              >>>>
              >>>>please write the chain of inequalities, it would be so much easier to
              >>>>visualize and understand.
              >>>>
              >>>>Thanks in advance,
              >>>>
              >>>>Maximilian
              >>>
              >>>
              >>>
              >>>
              >>>[Non-text portions of this message have been removed]
              >>>
              >>
              >
            • John
              You are not, after all, a bot, but afflicted and poleaxed by a non mobile Pollex. I suggest you get a Blackberry, and quickly, before they are banned.
              Message 6 of 6 , Aug 10, 2011
              • 0 Attachment
                You are not, after all, a bot, but afflicted and poleaxed by a non mobile Pollex. I suggest you get a Blackberry, and quickly, before they are banned.

                --- In primenumbers@yahoogroups.com, James Merickel <merk7777777@...> wrote:
                >
                >
                >
                > On Wed Aug 10th, 2011 4:22 PM EDT James Merickel wrote:
                >
                > >http://en.wikipedia.org/wiki/Concatenation_(mathematics) [cellphone chooses word I intend typing, and assumed parenthesis was typo]
                > >
                > >On Wed Aug 10th, 2011 3:07 PM EDT James Merickel wrote:
                > >
                > >>Another correction: The sentence claiming there are no 999-digit primes of other lengths than discussed should say 'with leading digit 9' as well.
                > >>Jim
                > >>
                > >>On Wed Aug 10th, 2011 3:00 PM EDT James Merickel wrote:
                > >>
                > >>>Sorry, link was missing open parenthesis. http://en.wikipedia.org/wiki/Concatenation_(mathematics)
                > >>>Jim
                > >>>
                > >>>----Forwarded Message----
                > >>>From: merk7777777@...
                > >>>To: primenumbers@yahoogroups.com
                > >>>Sent: Wed Aug 10th, 2011 2:55 PM EDT
                > >>>Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid
                > >>>
                > >>>Explicitly, using the notation of http://en.wikipedia.org/wiki/Concatenation_(mathematics) , the number I misread slightly is
                > >>>(10^142-519)||(10^142+1563)||(10^142+1083)||(10^142+301)||(10^142+13)||(10^142+111)||(10^142-563)
                > >>>
                > >>>----Forwarded Message----
                > >>>From: merk7777777@...
                > >>>To: primenumbers@yahoogroups.com
                > >>>Sent: Wed Aug 10th, 2011 2:39 PM EDT
                > >>>Subject: FW: Re: [PrimeNumbers] 999-digit prime factoid
                > >>>
                > >>>
                > >>>
                > >>>----Forwarded Message----
                > >>>From: merk7777777@...
                > >>>To: maximilian.hasler@...
                > >>>Sent: Wed Aug 10th, 2011 2:36 PM EDT
                > >>>Subject: Re: [PrimeNumbers] 999-digit prime factoid
                > >>>
                > >>>Correction placed in final sentence here.
                > >>>Three 333-digit consecutive primes are apt to exist such that their concatenation in some order is greatest of all non-titanic concatenations (Perhaps the specifics are interesting, but I didn't pursue it). Until n=9, then, there are no 999-digit prime concatenations of n consecutive primes other than the four I thought I described in (interesting enough) detail: Five must be 143 digits and the other two are those immediately under 10^142; and the reason I bothered with posting this is that three of these prime concatenations--the largest of them, I THOUGHT, placing all five of the ones with leading digit 1 left to right from largest to smallest, have the larger of the ones with leading digit 9 in initial position and the smaller one in final position. f still unclear, let me know. FWIW, I don't see much interesting in the seven primes involved other than that the second prime exceeding 10^142 does so by 111. In fact, THE FIRST IS ACTUALLY BEFORE
                > >>> THIS IN THE NUMBER THAT LOOKED INTERESTING TO ME, so I made a misstatement in posting this (Likely I would have thought it still worth a mention, but perhaps not). I can reproduce the numbers if actually wanted.
                > >>>Jim
                > >>>
                > >>>On Wed Aug 10th, 2011 11:26 AM EDT Maximilian Hasler wrote:
                > >>>
                > >>>>Dear Jim,
                > >>>>
                > >>>>for most mathematicians it takes quite some time to translate your
                > >>>>statements to formulae which give them a meaning.
                > >>>>
                > >>>>Could you be so kind as to use conventional mathematical language and
                > >>>>explicit formulae
                > >>>>instead (or at least in addition) to the "poetical" formulation ?
                > >>>>
                > >>>>below some concrete questions.
                > >>>>
                > >>>>On Wed, Aug 10, 2011 at 3:59 AM, James Merickel <merk7777777@...> wrote:
                > >>>>> Other than for concatenations of 3 of them,
                > >>>>
                > >>>>which three ?
                > >>>>
                > >>>>> the largest prime non-titanic concatenation of consecutive primes
                > >>>>
                > >>>>can you give it more explicitely ?
                > >>>>
                > >>>>> brackets
                > >>>>
                > >>>>how can ONE thing bracket something else ?
                > >>>>AFAIK you need two things to bracket something in-between !?
                > >>>>
                > >>>>> the five smallest
                > >>>>
                > >>>>could you give them ?
                > >>>>
                > >>>>> over 10^142
                > >>>>
                > >>>>what does this mean, "over" ?
                > >>>>
                > >>>>> in decreasing left-to-right order between the two largest under (also in decreasing order).
                > >>>>
                > >>>>the 2 largest under what ? can you give the values ?
                > >>>>
                > >>>>>  The next three largest are also concatenations of 7,
                > >>>>
                > >>>>what means " concatenations of 7 " ?
                > >>>>
                > >>>>> with the only one leading with the smallest
                > >>>>
                > >>>>what do you mean ?
                > >>>>
                > >>>>> having the second smallest in the 3rd position left-to-right.
                > >>>>
                > >>>>please write the chain of inequalities, it would be so much easier to
                > >>>>visualize and understand.
                > >>>>
                > >>>>Thanks in advance,
                > >>>>
                > >>>>Maximilian
                > >>>
                > >>>
                > >>>
                > >>>
                > >>>[Non-text portions of this message have been removed]
                > >>>
                > >>
                > >
                >
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