6 The polygonal root equals the number of points on the nth gnomon in the sequence of polygons counted from the initial point. In the terminology of the German Rechenmeister according to Faulhaber n had to be called the square root of the polygonal number and the nth term of the arithmetic progression, 1 + ( n – 1) d, was called the polygonal root. If the sequence of nested isosceles triangles is extended with a similar sequence like that on the right side of Figure 2, again starting from the left vertex, we can now read off the quadrilateral numbers from the base: 1, 4, 9, 16, etc. When d = 1 the successive triagonal numbers, 1, 3, 6, 10, etc., can be read off the base of the sequence of nested isosceles triangles depicted on the left side in Figure 2, starting from the left vertex. When d = 1 we get triangular (or triagonal) numbers when d = 2 we get quadrilateral (or tetragonal) numbers, etc. ![]() The Greek numeral that is used to denote the numbers corresponds exactly to d + 2. The names of polygonal numbers depend on d. Polygonal numbers are the sums of the first n terms of first order arithmetic sequences with first term 1 and difference d. It is conjectured, but as yet unproven, that there are infinitely many prime triples of each form. However, it is possible for p, p + 2, and p + 6 all to be prime, as in 5, 7, 11 or 17, 19, 23. To see this, note that if p leaves the remainder 1 when divided by 3, p + 2 is divisible by 3 whereas if p leaves the remainder 2, p + 4 is divisible by 3. ![]() ![]() It has been conjectured, but never proven, that there exist an infinite number of pairs of twin primes.Įxcept for the triplet (3, 5, 7), not all of the numbers p, p + 2, and p + 4 can be prime since one of them must be divisible by 3. Since 2 is the only even prime, for p > 2, consecutive primes must differ by two. Page, in Encyclopedia of Physical Science and Technology (Third Edition), 2003 I.H.1 Fundamental Theorem of ArithmeticĮvery natural number can be written uniquely as the product of primes.
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