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  • Problem from 'The Book of Squares'.

    From David Entwistle@qnivq.ragjvfgyr@ogvagrearg.pbz to rec.puzzles on Mon Sep 8 13:05:40 2025
    From Newsgroup: rec.puzzles

    In his introduction to 'The Book of Squares', Leonardo Pisano Fibonacci mentions a question. The author goes on to consider this question in Proposition 17. The question is:

    I wish to find a square number which increased or diminished by five
    yields a square number.

    In the comments on the proposition the translator, L. E. Sigler, notes
    the following: It is to be noted that the congruent square solutions
    sought by Leonardo in this theorem are not whole numbers, but fractions.

    This will be the basis of the previous question on squares in arithmetic progression covered in Dudeney's puzzle 128. - A Problem in Squares.
    --
    David Entwistle

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  • From James Dow Allen@user4353@newsgrouper.org.invalid to rec.puzzles on Tue Sep 9 05:25:53 2025
    From Newsgroup: rec.puzzles


    David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> posted:

    In his introduction to 'The Book of Squares', Leonardo Pisano Fibonacci mentions a question. The author goes on to consider this question in Proposition 17. The question is:

    I wish to find a square number which increased or diminished by five
    yields a square number.

    In the comments on the proposition the translator, L. E. Sigler, notes
    the following: It is to be noted that the congruent square solutions
    sought by Leonardo in this theorem are not whole numbers, but fractions.

    This will be the basis of the previous question on squares in arithmetic progression covered in Dudeney's puzzle 128. - A Problem in Squares.


    A factional number which provides satisfaction is
    1681/144 = (41/12)^2
    Increasing by five (720/144) yields another square
    1681/144 + 720/144 = (49/12)^2
    Decreasing by five also yields a square
    1681/144 - 720/144 = (31/12)^2

    This solution presented itself when I substituted x=5, y=4 into
    Leonardo's Congruum Theorem.
    (See https://fabpedigree.com/james/fibflt4.htm)
    That theorem deals with the exact question asked here:
    When are three perfect squares in arithmetic progression?

    A list of "greatest mathematicians" can be found on the 'Net
    which places Leonardo Pisano Fibonacci in the #32 slot.
    This 13th-century Italian is famous for several things but his
    Congruum Theorem is a highlight of his creative mathematics.
    I think that Leonardo's Proposition XVI in his Book of Squares
    (Liber Quadratorum) should be credited as a proof of the N=4
    case of Fermat's Last Theorem!

    Cheers,
    James
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  • From James Dow Allen@user4353@newsgrouper.org.invalid to rec.puzzles on Tue Sep 9 05:37:12 2025
    From Newsgroup: rec.puzzles


    James Dow Allen <user4353@newsgrouper.org.invalid> posted:

    This solution presented itself when I substituted x=5, y=4 into
    Leonardo's Congruum Theorem.
    (See https://fabpedigree.com/james/fibflt4.htm )

    Oops. I forgot to leave a space after the URL to facilitate clicking.

    C,
    J
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  • From Richard Heathfield@rjh@cpax.org.uk to rec.puzzles on Tue Sep 9 07:29:42 2025
    From Newsgroup: rec.puzzles

    On 09/09/2025 06:37, James Dow Allen wrote:

    James Dow Allen <user4353@newsgrouper.org.invalid> posted:

    This solution presented itself when I substituted x=5, y=4 into
    Leonardo's Congruum Theorem.
    (See https://fabpedigree.com/james/fibflt4.htm )

    Oops. I forgot to leave a space after the URL to facilitate clicking.

    RFC1738 says: The characters "<" and ">" are unsafe because they
    are used as the delimiters around URLs in free text

    RFC1738 is updated by RFC3986, but RFC3986 confirms: "Using <>
    angle brackets around each URI is especially recommended as a
    delimiting style".

    Thus a browser should make

    See this URL<https://fabpedigree.com/james/fibflt4.htm>right here

    clickable.
    --
    Richard Heathfield
    Email: rjh at cpax dot org dot uk
    "Usenet is a strange place" - dmr 29 July 1999
    Sig line 4 vacant - apply within
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  • From David Entwistle@qnivq.ragjvfgyr@ogvagrearg.pbz to rec.puzzles on Wed Sep 10 09:46:50 2025
    From Newsgroup: rec.puzzles

    On 09/09/2025 06:25, James Dow Allen wrote:
    I think that Leonardo's Proposition XVI in his Book of Squares
    (Liber Quadratorum) should be credited as a proof of the N=4
    case of Fermat's Last Theorem!

    Proposition 16

    "I wish to find a congruous number which is a square multiple of five".

    'Congruous number' isn't an expression I'm familiar with, but Wikipedia
    says of congruent numbers:

    "In number theory, a congruent number is a positive integer that is the
    area of a right triangle with three rational number sides. A more
    general definition includes all positive rational numbers with this
    property".

    That page mentions the work of Leonardo, so I assume it is the same topic.
    --
    David Entwistle
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  • From Charlie Roberts@croberts@gmail.com to rec.puzzles on Wed Sep 10 16:21:00 2025
    From Newsgroup: rec.puzzles

    On Mon, 8 Sep 2025 13:05:40 +0100, David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

    I wish to find a square number which increased or diminished by five
    yields a square numbe

    Just saw that thread and having missed the last step in the
    divisor puzzle (where Dudeny's comment about there being
    a great shortcut misled me), I wonder what other
    restrictions are put for the above problem. If one is
    willing to wander off into to complex plane would

    -1, 4, 9

    work? That is, will 4 be an acceptable answer?

    However, if the question is couched with Number Theory
    in mind, that above is pure nonsense!


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