-------------------- 1-5 2,3,5,7,13 in or before the middle ages 6-7 17,19 1588 Cataldi 8 31 1750 Euler 9 61 1883 Pervouchine 10 89 1911 Powers 11 107 1914 Powers 12 127 1876 Lucas 13-14 521,607 1952 Robinson 15-17 1279,2203,2281 1952 Lehmer 18 3217 1957 Riesel 19-20 4253,4423 1961 Hurwitz & Selfridge 21-23 9689,9941,11213 1963 Gillies 24 19937 1971 Tuckerman 25 21701 1978 Noll & Nickel 26 23209 1979 Noll 27 44497 1979 Slowinski & Nelson 28 86243 1982 Slowinski 29 110503 1988 Colquitt & Welsh jr. 30 132049 1983 Slowinski 31 216091 1985 Slowinski 32? 756839 1992 Slowinski & Gage 33? 859433 1994 Slowinski and Gage,
Figurate Numbers. Numbers geometrically constructed had a particular importance to the Pythagorean s.
Triangular numbers. These numbers are 1, 3, 6, 10, ... . The general form is the familiar
Square numbers These numbers are clearly the squares of the integers 1, 4, 9, 16, and so on. Represented by a square of dots, they prove(?) the well known formula
The gnomon is basically an architect's template that marks off "similar" shapes.
Note the gnomon has been placed so that at each step, the next odd number of dots is placed.
Figurate Numbers of any kind can be calculated.
The Pentagonal and Hexagonal numbers are shown in the graphs below.
Note that the sequences have sums given by
Similarly, polygonal numbers of all orders are dsignated; this process can be extended to three dimensional space, where there results the polyhedral numbers. Philolaus is reported to have said:
All things which can be known have number; for it is not possible that without number anything can be either conceived or known.
The Pythagorean Theorem
Whether Pythagoras learned about the 3, 4, 5 right triangle while he studied in Egypt or not, he was certainly aware of it. This fact though could not but strengthen his conviction that all is number. It would also have led to his attempt to find other forms. How might he have done this?
One place to start would be with the square numbers, and arrange that three consecutive numbers be a Pythagorean triple! Consider for any odd number m,
which is the same as
Put the gnomon around . The next number is 2n+1, which we suppose to be a square.
It follows that
This idea evolved over the years and took other forms.
Did Pythagoras or the Pythagorean s actually prove the Pythagoras Theorem? Probably not. Although a proof is simple to give, the Pythagorean s had only a limited theory of similarity. And perhaps the reason was that rigor had not yet advanced to that level, at least in the early and middle period. The late Pythagorean s ( 400 B.C) however probably did supply a proof of this most famous of theorems.
Theorem I-47. In right-angled triangles, the square upon the hypotenuse is equal to the sum of the squares upon the legs.
This figure is modelled after the original figure used by Euclid to prove the result. It is known to the French as pon asinorum and to the Arabs as the Figure of the Bride.
More Pythagorean Geometry
Contributions by the Pythagorean s include
From Kepler we have the quote
``Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."
A line AC divided into extreme and mean ratio is defined to mean that it is divided into two parts, AP and PC so that AP:AC=PC:AP, where AP is the longer part.
Let AP=x and AC= a. Then the golden section is
and this gives the quadratic equation
The solution is
The golden section is the positive root:
And this was all connected with the construction of a pentagon.
The Pythagorean Pentagram
First we need to construct the golden section. The geometric construction, the only kind accepted, is illustrated below.
Assume the square ABCE has side length a. Bisecting DC at E construct the diagonal AE, and extend the segment ED to EF, so that EF=AE. Construct the square DFGH. The line AHD is divided into extrema and mean ratio.
The key to the compass and ruler construction of the pentagon is the construct