Platos hidden theorem on the distribution of primes

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Platos hidden theorem on thedistribution of
primes

• Antonis Vardulakis and Georgios Velisaris
• Department of Mathematics
• Aristotle University of Thessaloniki
• Thessaloniki 54006, Greece

Clive Pugh and Peter Shiu Department of
Mathematics, University of Loughborough Loughborou
gh, U.K.
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Abstract

• The story behind a conjecture by the late
  professor Andreas Zachariou of the Department of
  Mathematics of the University of Athens that a
  passage in Book 5, 737e, 738 of Plato's "Laws" is
  in fact a hidden theorem concerning the
  arrangement of prime numbers is revisited and two
  independent proofs of this remarkable result are
  given.

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The Laws is Platos last and longest dialogue.
Plato The Laws NOMOI
It is generally agreed that Plato wrote this
dialogue as an old man, having failed in his
effort in Syracuse on the island of Sicily to
guide a tyrant's rule, instead having been thrown
in prison.
Born. 428-427 BC, Athens Died. 348-347 BC, Athens
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The setting of the Laws.

• Unlike most of Plato's dialogues, Socrates does
  not appear in the Laws.
• This is fitting because the dialogue takes place
  on the island of Crete, and Socrates never
  appears outside of Athens in Plato's writings,
  except in the Phaedrus, where he is just outside
  the city's walls.
• In the Laws instead of Socrates we have the
  Athenian Stranger (in Greek, Xenos - ?????) and
  two other old men, an ordinary Spartan citizen
  (Megillos) and a Cretan politician and lawgiver
  (Kleinias) from Knossos.
• The Athenian Stranger, who is much like Socrates
  but whose name is never given, joins the other
  two on their religious pilgrimage to the
• cave of Zeus on the Mount Ida on the Island of
  Crete.
• The entire dialogue takes place during this
  journey, which mimics the action of Minos, who is
  said by the Cretans to have made their ancient
  laws, who walked this path every nine years in
  order to receive instruction from Zeus on
  lawgiving.

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Before the Mathematics some Cretan Geography and
Greek Mythology and Etymology associated with the
Laws.
Possible path taken by the three friends in the
Laws on their religious pilgrimage to Zeus
birth place
My village (Vardulakis) of Anidri where in 2003,
the Anidri meeting of some friends
Mathematicians and Engineers took place and the
Zachariou conjecture on Platos hidden
theorem was presented to the participants asking
everybody to try to prove it.
Idaion Andron (?da??? ??t???) (Cave of Zeus,
his birth place on Mount Ida)
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Mythology-Geography-Etymology????????a-Ge???af?a-
?t?µ?????a

• Platos dialogue takes place at the Island of
  Crete during a journey by the three friends to a
  place of pilgrimage at the cave of Zeus, at 1530
  meters, on the Mount of Ida named Idaion Andron
  (?da??? ??t??? ) (Cave of Mount Ida (?d?).
• According to the Greek Mythology, Zeus, the
  father of all Gods, was born by Rea who sheltered
  her newly born child in the cave to save it from
  her husbant Kronos ?????? (Time ) (Saturn)
  who used to eat his children.
• Despite all that Kronos Xronos Time, still to
  this day, consumes the rest of us (its
  children).

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Saturn Devouring One of His Sons, by Francisco
Goya
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Background Mythology - Etymology

• Hidden in this cave, Zeus was protected and
  raised by a goat named Amalthea, (?µ???e?a
  tender goddess) drinking her milk. Her horns,
  the so called horns of plenty, to this day
  symbolize all the goodies (mediteranean diet)
  needed to feed a baby that will grow up to become
  the father of all Gods and be protected for ever
  from being consumed by Kronos ?????? Time.
• In Greek a goat is called ????? Aegis .
• That is why when we say in Greek (and in English)
  ?p? t?? ????da Under the Aegis, (which
  literally means under the goat), we mean
  under the protection.
• Also, a rainy storm, in modern Greek is called
  
• ?at-a???da under the goat. This is
  because, Zeus, on stormy days up in Mount Ida,
  used to find shelter under the Amalthea goat.
• According to another tradition Amalthea's
  skin, killed and skinned by the grown Zeus,
  became his protective shield aegis.

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???S ZEUS
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Back to Platos Laws

• In Book 5, 737e, 738 of Plato's "Laws" it is
  stated that the number of citizens of an ideal
  City State should be 5040 because this number its
  divisible by a total of 59 numbers and in
  particular by all integers from 1 up to 10.

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The page in Platos Laws where the number 5040
and its remarkable properties are first mentioned
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An English translation of the part of book 5 of
Plato's Laws where the number 5040 is mentioned

• " 737e Let us assume that there are--as a
  suitable number--5040 men, to be land-holders and
  to defend their plots2 and let the land and
  houses be likewise divided into the same number
  of parts--the man and his allotment forming
  together one division. man who is making laws
  must understand at least thus much,-- First, let
  the whole number be divided into two next into
  three then follow in natural order four and
  five, and so on up to ten. Regarding numbers,
  every 738a what number and what kind of number
  will be most useful for all States. Let us choose
  that which contains the most numerous and most
  consecutive sub-divisions. Number as a whole
  comprises every division for all purposes
  whereas the number 5040, for purposes of war, and
  in peace for all purposes connected with
  contributions and distributions, will admit of
  division 738b into no more than 59 sections,
  these being consecutive from one up to ten.

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The number 5040 according to Wikipedia
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THE ZAHARIOU CONJECTURE
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The Proof

• Although until 2003 the conjecture had been
  tested to be true for very big successive primes,
  to our knowledge, up to the summer of 2003, no
  proof of the Theorem was available.
• The first proof was given by Peter Shiu in 2004
  after the conjecture was mentioned to him by C.
  Pugh.
• The second proof was given by a Medicine
  undergraduate, Georgios Velisaris, in 2007.

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Fori 1, i lt 1000, n Primei m Primei
1 xn n! apot True Forj n 1, j lt
m, apot apot (Modn!, j
0) j Print"n", n, " m", m, "
"apot IfNotapot, Print"Ahhhhhhhhhhhhhhhh"
i
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Plato's sieve for the primes
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Conclusions

• A conjecture by the late professor Andreas
  Zachariou that a passage in Book 5, 737e, 738 of
  Plato's "Laws" constitutes a theorem concerning
  the arrangement of prime numbers has been proved
  to be true.
• Finally as Werner Heisenberg once said
• I think that modern physics has definitely
  decided in favour of Plato. In fact the smallest
  units of matter are not physical objects in the
  ordinary sense they are forms, ideas which can
  be expressed unambiguously only in mathematical
  language.
• I recommend every body to

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