![]() His claims to be able to reveal the intentions of God by means of the interpretation of the biblical numbers became explicit, at the latest, with his Newer Mathematischer Kunstspiegel (The New Artistic Mirror of Mathematics). The fact that the great mathematicians and philosophers of antiquity did not know anything about the biblical numbers, or at least did not express themselves on the interpretation of them, was understood by Faulhaber as an indication of God's intention to hide such information until his time. 9 Invariably the seven biblical numbers 2300, 1290, 1335, 666, 1260, 16, repeatedly interpreted figuratively, played a special role. He applied various methods, among others a calculus of differences to get his formulae. This led to arithmetic progressions of higher order and their sums, in particular sums of powers of natural numbers up to the exponent 17, for which he gave general formulae. In later works, in particular in his Miracula Arithmetica of 1622 7 and in his Academia Algebrae of 1631, 8 starting from figurate numbers like polygonal numbers and pyramidal numbers, Faulhaber introduced other solid numbers, like prismatic or dodecahedronal numbers. 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. When dealing with sequences, we use \(a_n\) in place of \(y\) and \(n\) in place of \(x\).Robert L. Recall the slope-intercept form of a line is \(y=mx b\). ![]()
0 Comments
Leave a Reply. |