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PYTHAGOREAN TRIPLES
7-11-13 * 11-13-17 * 41-53-67
(Wrong Phytagorean Triplets consisted of 3 naturals odd prime numbers whereas necessarily one on the sides of the right angle must be even).
PYTHAGORAS
Pythagoras many centuries after the pyramids will
state the relation between the sides of the particular
right-angled triangle whose dimensions are respectively
equal to 3,4 and 5, nowadays definite as follows:
A pythagorician's continuation is a continuation of natural nonnull numbers
(x, y, z)which check the relation: x² + y² = z²
There is an infinity of them.
The mathematical research on this subject is a constant since antiquity:
4000 years B.C. Babyloneans had a mathematical knowledge,
and this question of the triples is already treated in the shelf
babylonean wedge-shaped known as Plimpton 322 written towards -1900,
it will be continued by Thalès and Pythagore -500 B.C., then by Euclide,
Diophante d'Aléxandrie and even more close to us by Pierre de
Fermat,Euler, Gauss etc..
That's The continuum of the mathematics'adventure.
The triangle known as of Pythagoras is present on
several occasions in the constructions and is not the result
of the chance. . .
However Ancient Egypt don't know the multiplication nor
division, consequently the squares and the square roots.
This knowledge for example could be issued from a systematic
research or be the consequence of the implementation of
particular values.
What about 7 and 11?
The one who is for example a little interested in the
pyramid of Cheops knows or could note that its
proportions result from the ratio 7/11. (Height 280 cubits
either 7x40 and bases 440 cubits, or 11x40).
These are two prime numbers which are following and which
generate a identical property to the triangle of
Pythagoras:7x7=49 plus 11x11=121 equal 170.
However the square root of 170 is 13,0384 for the
hypotenuse.
i.e. with the hundredth precision a triangle which is unique
using a prime numbers'continuation 7-11-13 .
Does there exist others?
I am not informed of publication or unspecified evocation
on this subject, and I met only three mathematical
continuations of prime numbers in this case when I had
worked out a calculation programme for my computer through
the first 15.000 prime numbers and this with continuations
consisted of progressive intervals going from 1 to 500.
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7-11-13 (hypotenuse: 13,0384)that is three prime
numbers which are following, (4/100èmes Error),
11-13-17 (hypotenuse: 17,0294) idem above,
(3/100èmes Error),
41-53-67 (hypotenuse: 67,007) that is also three increasing
prime numbers separated by two others,
(7/1000èmes Error),
and there is no other:
(Note) the simple fact of taking a multiple (triangle
proportional) of these values will lead that we will not use
any more prime numbers, or they will not be integers.
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If these results are not in strict agreement with our current mathematical rules 2.700 years B.C.
it could probably be due to a
research using only the physical measurements which could not
have an absolute precision as our current calculators
and the results by the mean of drawings were certainly considered as correct.
Only the three evoked
triangles generate a continuation of the Pythagoras'
type.
If one refers to the use of the 5 prime numbers, to a
systematic research, and to the fact that the multiplication
and division were not known, a fortiori the square roots,
this choice of 7/11 for Cheops could not be
fortuitous.
The paving of the King's Chamber 
has several answers to these questions.
Continuing my research, it appeared to me that 7-11-13
was dedicated to Cheops each one having seen like
an obviousness the ratio 7/11 of the pyramid. (Height 280,
Bases 440).
Its base can be also written 10 times 7+11+13 plus 10 times 13, (i.e:440).
Concerning Chephren, it is less obvious and I have
few elements about it except its dimensions
It is probable that the mathematical continuation
11,13,17 is dedicated to Chephren;
One can simply note that its base is equal to 10 times
the sum of 11+13+17, (i.e:410).
These particular triangles of Pythagoras' type were well
known and present as Mycerinus whose measurements are base=200,
height=125, i.e. using two triads 3-4-5 (75-100-125).
Moreover these so particular and single triangles have other
properties which we can only note, and for example:
Note: (In the current values used in Giza..)
In fact there are 3 differents before the prime number 4999:
229-307-383 (hypotenuse: 383,0013)that is also three increasing prime numbers separated by two others.
491-643-809 (hypotenuse: 809,0303)that is also three increasing prime numbers separated by one other.
571-743-937 (hypotenuse: 937,0646)that is also three increasing prime numbers separated by five others.
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