First Communication


Strange things in a mathematical train

1) Pi is not a innocent number. It seems to me like a infinite train whose wagons carries, from time to time, some attractive surprises. For instance, we can found in it a certain sympathy for the number 5: the first 5 decimals sums 20, the first 20 sums 5x20=100. Curiously, the first 100 sums 477, a number instrinsically connected with the geometry of the Great Pyramid, as we'll see later.

More remarkable is what happens to the 761st decimal. It is a 4 followed by 6 nines (a very rare sequence) that gives it a virtual value of 5, which is also the digital root of 761.

Now let's do: 7!+6!+1!=5761. We obtain the same number preceded by its digital root. And 761 is the unique number that gives such a result, according to the equation

(the problem has a limit: n<=7, and it was resolved by an adequate computer program by eng. Jorge Cruz. Solution later confirmed analitically by eng. José Cayolla).

So, we can see in Pi a rarity that points out a singularity. I've called it "The Singularity of Pi". (see: Carlos Calvet, "Matemática e Simbologia" in Colóquio-Artes nr. 85, June. 90, edited by Calouste Gulbenkian Foundation, Lisbon, Portugal).

2) Number 761 gives us another surprise, in the context of a mathematical rule that says:

"When the sum of 2 different numbers is 1,000 and one of them has digital root of 5, the other will have identical digital root. This only happens with number 1,000".

There are only 110 numbers in those conditions, forming 55 pairs. Of those, 9 pairs are formed by prime numbers. All those numbers differ by 9 or by multiples of 9.

From the 55 pairs, and situated in the middle of the conjunct, only the pair formed by 761 and 239 (prime numbers) is solution to the equation

So we have: 5761/761-6=5(239/761)=1.5703022

This constitutes another singularity of the 761.

Note that 10(239/761)=3.1406… the sole from the 55 pairs that gives the closest approximation to Pi, with 3 digits correct.

There is another pair: 500-500, but formed by the same number; also it don't satisfy the equation. In the near future I'll give the complete list of the 55+1 pairs.

3) If we do: Pi=22/7=3.142857 142857 … whose decimals are composed by periodic sequences of 6 numbers, we can verify that the 761st decimal is a 5 and it appears in the 5th place of the 127th sequence; 127=2^7-1 is the 7th number of Mersenne.
Note, by the way, as a curiosity, that the inverse of 22/7 gives the velocity of light:

only 19 Km/s slower than the theoric value (perhaps shall it be the velocity of light slowed down by the cosmic dust…!).

4) Pi has too curious relations with 3 and 7:

  • The first 3 decimals sums 6, the first perfect number and the triangular number of 3.
  • The first 7 decimals sums 28, the second perfect number and the triangular number of 7.

  • 5) Another curious relation with 7: the 9961st decimal of Pi is a 6 followed by 3 nines, that gives it a virtual value of 7. Or 9961 is divisible by 7 (as 5761) and its digital root is 7. Otherwise 9!+9!+6!+1!=726481, also divisible by 7. And if we do: 9961-5761=4200=7x6x1x100. (we can see Pi with 10,000 decimals in Petr Beckmann's A history of Pi, The Golem Press, Boulder, Colorado, U.S.A., 1982; or in Mathematics of computation, The American Mathematical Society, 1962, vol. XVI, nr. 77, pp 76-99).

    N.B. -

    6) Another notable event is the relation of Pi to the magic square of 5 (found by the American E. Lobeck, ca. 1975). Here is a brief description:

    Consider the magic square of 5 whose rows are:

    1st row: 17, 24, 1, 8, 15;
    2nd row 23, 5, 7, 14, 16;
    3rd row 4, 6, 13, 20, 22;
    4th row 10, 12, 19, 21, 3
    5th row 11, 18, 25, 2, 9.

    Its constant is 65, the sum of each row, column or diagonal.

    Substituing successively in that square the numbers 1, 2, 3, …25, by the first 25 digits of Pi (3, 1, 4, 1, 5, 9 …) we obtain a new square whose sums of columns and rows are identical, 2 by 2. That is to say:

    1st column=17=5th row;
    2nd column=29=4th row;
    3rd column=25=3rd row;
    4th column=24=1st row;
    5th column=23=2nd row

    Somewhat misterious, isn't it?

    (see: Lucien Gérardin, Les carrés magiques, p.167, ed. Dangles, 1985, St. Jean-de-Braye, France; or "Coloquio Artes" nr. 85, p. 15, above mentioned).

    Also the sum of the two diagonals is 65 (27+38), the constant of the magic square of 5. In this square the crossing of the diagonals is number 13. Curiously, the first 13 decimals of Pi sums 5x13=65.

    7) A brief commentary: the "Singularity of Pi" set us a question - is it a casual coincidence? It seems to me too much casual. What shall it be then: a signal of an occult structure?

    I leave this interrogation hovering in the cyberspace...
    (to be continued)

    Carlos Calvet
    Lisbon, 6-May-1997