Pythagoras was born on the island of Samos, Greece, in 569 BC.He excelled as a student and, as a young man, he traveled widely . Tradition says that he explored from India in the East to Gaul in the West.Pythagoras traveled extensively through Egypt, learning maths, astronomy and music. Pythagoras left Samos in disgust for its ruler Polycrates. He moved on to the Greek city of Crotona, located on the southern shore of Italy. There he created a school where his followers lived and worked.It was a mystical learning community.

At the heart of Pythagoras` teachings was the vision of the underlying harmony of the universe. This harmony had to be abstracted from the confusion of visible things and daily events. As a matter of fact this harmony existed in the abstract - in the same way as numbers and mathematical formulas are abstractions.
Pythagoras believed in secrecy and communalism, so it is almost impossible distguishing his work from the work of his followers. Pythagoras and his followers contributed to music, astronomy and mathematics. He died about 500 BC in Metapontum, Lucania.

Pythagoras` desire was to find the mathematical harmonies of all things. The study of of odd, even, prime and square numbers were among numerous mathematical investigations of the Pythagoreans. This helped them develop a basic understanding of mathematics and geometry to build their Pythagorean theorem.
The Pythagorean Theorem states that the square of the hipotenyse of a right triangle is equal to the sum of the squares of the other two sides.

Pythagoras Theorem asserts that for a right triangle with the short sides of the length a and b and the long side of the length c a2 + b2 = c2

A Pythagorean Triangle is a right angled triangle which sides are whole numbers. There is an easy way to generate Pythagorean triangles using 4 Fibonacci numbers. Take, for example, the 4 Fibonacci numbers 1,2,3 and 5 .
We can now make a Pythagorean Triangle as follows :

     

Mutiply the two middle or inner numbers

here 2 and 3 giving 6;

Double the results

here twice 6 gives 12

That is one side, a, of the Pythagorean Triangle:

a=2*F(n+1)*F(n+2)

Multiply together the two outer numbers

here 1 and 5 giving 5.

This is the second side, b, of the Pythagorean Triangle:

b=F(n)*F(n+3)

The third side, the longest one, is found by adding together the squares of the inner

two numbers

here 2²=4 and 3²=9 and their sum is 4+9=13.

This is the third side, c, of the Pythagorean Triangle:

c=F²(n+1) + F²(n+2)

     

We have generated 12, 5, 13 Pythagorean Triangle.

     

n	F(n)	F(n+1)	F(n+2)	F(n+3)	a	b	c
1	1	1	2	3	4	3	5
2	1	2	3	5	12	5	13
3	2	3	5	8	30	16	34
4	3	5	8	13	80	39	89
6	8	13	21	34	546	272	610
7	13	21	34	55	1428	715	1597
8	21	34	55	89	3740	1869	4181
9	34	55	89	144	9790	4869	10946

     

In fact, this process works not only for Fibonacci numbers but for Lucas numbers as well.

     

n	L(n)	L(n+1)	L(n+2)	L(n+3)	a	b	c
1	1	3	4	7	24	7	25
2	3	4	7	11	56	33	65
3	4	7	11	18	154	72	170
4	7	11	18	29	396	203	445
5	11	18	29	47	1044	517	1165
6	18	29	47	76	2726	1368	3050
7	29	47	76	123	7144	3567	7985
8	47	76	123	199	18696	9353	20905
9	76	123	199	322	48954	24472	54730

     

However, for a Pythagorean Triangle, we also want the sides to be Fibonacci numbers too.
It is triangle with sides
a=F(n), b=F(n+1) and c=sqrt(F(2n+1));
Here is a list of some Pythagorean triangles with sides which are Fibonacci numbers :

     

n	F2(n)	F2(n+1)	F(2*n+1)
1	1	1	2
2	1	4	5
3	4	9	13
4	9	25	34
5	25	64	89
6	64	169	175
7	169	441	610
8	441	1156	1597
9	1156	3025	4181

     

where

     

F²(n) + F²(n+1) = F(2*n+1)

     

Also, a Pythagorean Triangle can be a right angled triangle with sides which are Lucas numbers:
a=L(n), b=L(n+1) and c= sqrt( 5*L(2n+1))
Here is a list of some Pythagorean triangles with sides which are Lucas numbers:

     

n	L2(n)	L2(n+1)	5*F(2*n+1)
1	1	9	10
2	9	16	25
3	16	49	65
4	49	121	170
5	121	324	445
6	324	841	1165
7	841	2209	3050
8	2209	5776	7985
9	5776	15129	7305

     

where

     

L²(n) + L²(n+1) = 5*F(2*n+1)

     

or

     

     

By adding the square of Fibonacci ( or Lucas ) numbers we obtain many interesting relations:

     

F²(n) + 2*F²(n+1) + F²(n+2)= L(2*n+2)

L²(n) + 2*L²(n+1) + L²(n+2) = 5*L(2*n+2)

     

and

     

F²(n) + 3*F²(n+1) + 3*F²(n+2) + F²(n+3) = 5*F(2*n+3)

L²(n) + 3*L²(n+1) + 3*L²(n+2) + L²(n+3)= 25*F(2*n+3)

     

and

     

F²(n)+4F²(n+1)+6F²(n+2)+4F²(n+3)+F²(n+4)=5L(2n+4)

L²(n)+4L²(n+1)+6L²(n+2)+4L²(n+3)+L²(n+4)=25L(2n+4)

     

and

     

F²(n)+5F²(n+1)+10F²(n+2)+10F²(n+3)+5F²(n+4)+F²(n+5)=25F(2n+5)

L²(n)+5L²(n+1)+10L²(n+2)+10L²(n+3)+5L²(n+4)+L²(n+5)=125F(2n+5)

     

and so on...

     

     

or

     

     

In general case we have the next relation between Lucas and Fibonacci numbers:

     

See Also:  Fibonacci Numbers in Nature  Fibonacci Numbers and the Pascal Triangle   FIBONACCI IDENTITIES FROM MATRICES AND VECTORS    Fibonacci Numbers and The Natural Logarithmic Base, e   Archimedes` constant PI and the Fibonacci Numbers

     

     

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