Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

In this article we shall examine several different types of figurative numbers which have been studied extensively over the period of 2500 years, and currently scattered on hundreds of websites. We shall discuss their computation through simple recurrence relations, patterns and properties, and mutual relationships which have led to curious results in the field of elementary number theory. Further, for each type of figurative numbers we shall show that the addition of first finite numbers and infinite addition of their inverses often require new/strange techniques. We sincerely hope that besides experts, students and teachers of mathematics will also be benefited with this article.


Introduction
Pythagoras of Samos (around 582-481 BC, Greece) and his several followers, especially, Hypsicles of Alexandria (around 190-120 BC, Greece), Plutarch of Chaeronea (around 46-120, Greece), Nicomachus of Gerasa (around 60-120, Jordan-Israel), and Theon of Smyrna (70-135, Greece) portrayed natural numbers in orderly geometrical configuration of points/dots/pebbles and labeled them as figurative numbers. From these arrangements, they deduced some astonishing number-theoretic results. This was indeed the beginning of the number theory, and an attempt to relate geometry with arithmetic. Nicomachus in his book, see [1], originally written about 100 A.D., collected earlier works of Pythagoreans on natural numbers, and presented cubic figurative numbers (solid numbers). Thus, figurate numbers had been studied by the ancient Greeks for polygonal numbers, pyramidal numbers, and cubes. The connection between regular geometric figures and the corresponding sequences of figurative numbers was profoundly significant in Plato's science, after Plato of Athens (around 427-347 BC, Greece), for example in his work Timaeus. The study of figurative numbers was further advanced by Diophantus of Alexandria (about 250, Greece). His main interest was in figurate numbers based on the Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron), which he documented in De solidorum elementis. However, this treatise was lost, and rediscovered only in 1860. Dicuilus (flourished 825, Ireland) wrote Astronomical Treatise in Latin about 814-816, which contains a chapter on triangular and square numbers, see Ross and Knott [2]. After Diophantus's work, several prominent ma-  [3] had given an extensive information about figurative numbers.
In this article we shall systematically discuss most popular polygonal, centered polygonal, three dimensional numbers (including pyramidal numbers), and four dimensional figurative numbers. We shall begin with triangular numbers and end this article with pentatope numbers. For each type of polygonal figurative numbers, we shall provide definition in terms of a sequence, possible sketch, explicit formula, possible relations within the class of numbers through simple recurrence relations, properties of these numbers, generating function, sum of first finite numbers, sum of all their inverses, and relations with other types of polygonal figurative numbers. For each other type of figurative numbers mainly we shall furnish definition in terms of a sequence, possible sketch, explicit formula, generating function, sum of first finite numbers, and sum of all their inverses. The study of figurative numbers is interesting in its own sack, and often these numbers occur in real world situations. We sincerely hope after reading this ar-ticle it will be possible to find new representations, patterns, relations with other types of popular numbers which are not discussed here, extensions, and real applications.

Triangular Numbers
In this arrangement rows contain 1, 2,3, 4, , n  dots (see Figure 1). From Figure 1 it follows that each new triangular number is obtained from the previous triangular number by adding another row containing one more dot than the previous row added, and hence n t is the sum of the first n positive integers, i.e., i.e., the differences between successive triangular numbers produce the sequence of natural numbers. To find the sum in (1) we shall discuss two methods which are innovative.
Young Gauss found the Formula (2) instantly and wrote down the correct answer 5050.
Method 2. From Figure 2 Proof without words of (2) is immediate, see Alsina and Nelsen [4]. However, a needless explanation is a "stairstep" configuration made up of one block plus two blocks plus three blocks, etc, replicated it as the shaded section in Figure 2, and fit them together to form an ( )  representing n t ) and the rectangle's area is the product of base and height, that is, ( ) 1 n n + , then the stairstep's area must be half of the rectangle's, and hence (2) holds.
To prove (2) (1), and (4) becomes same as (2). From (4), it is also clear that Sd a d a n d He also provided elegant results for the summation of series of squares and cubes. In Rhind Papyruses (about 1850 and 1650 BC) out of 87 problems two problems deal with arithmetical progressions and seem to indicate that Egyptian scriber Ahmes (around 1680-1620 BC) knew how to sum such series. For example, Problem 40 concerns an arithmetic progression of five terms. It states: divide 100 loaves among 5 men so that the sum of the three largest shares is 7  ). There is a discussion of arithmetical progression in the works of Archimedes of Syracuse is known for the distribution of prime numbers in arithmetic progressions. Terence Chi-Shen Tao (born 1975, Australia-USA) showed that there exist arbitrarily long arithmetic progressions of prime numbers.
The following equalities between triangular numbers can be proved rather easily.
provided a/d is nonpositive.  We shall show that for an integer 1 k > , ( ) mod , 1 n t k n ≥ repeats every k steps if k is odd, and every 2k steps if k is even, i.e., if  is the smallest positive integer such that for all integers n ( )( ) ( ) ( ) Now if k is odd, then in view of ( ) 1  This implies that k ≥  , because  is the smallest integer for which (8) holds. But, then from (9) it follows that k =  .
For example, for ( )   It is known, see Trigg [9], that an infinity of palindromic triangular numbers exist in several different bases, for example, three, five, and nine; however, no infinite sequence of such numbers has been found in base ten.
 Let m be a given natural number, then it is n-th triangular number, i.e., it follows that if n is the sum of two triangular numbers, then 4 1 n + is a sum of two squares.
is the generating function of all triangular numbers. In 1995, Sloane and Plouffe [10] have shown that To find the sum of the first n triangular numbers, we need an expression for 2 1 n k k = ∑ (a general reference for the summation of series is Davis [11]). For this, we begin with Pascal's identity ( ) From (13) and (14), the Formula (12) follows. Another proof of (11)   On adding these n equations and cancelling the common terms, (11) follows. Now from (2) and (11) Relation (15) is due to Aryabhata.
For an alternative proof of (15), we note that , i.e., three successive triangular numbers whose sum is a perfect square. Similarly, we have From (15), we also have  The reciprocal of the ( ) and hence Jacob Bernoulli (1654-1705, Switzerland) in 1689 summed numerous convergent series, the above is one of the examples. In the literature this procedure is now called telescoping, also see Lesko [12].  Pythagoras theorem states that if a and b are the lengths of the two legs of a right triangle and c is the length of the hypothenuse, then the sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse, i.e., A set of three positive integers a, b and c which satisfy (18)  Plato asserted that the creation is perfect because the number 6 is perfect.
They also realized that like squares, six equilateral triangles (see We note that if ( ) ), it follows that the iterative scheme 1 1 gives all solutions of 2 System (20) can be written as This result is originally due to Euler which he obtained in 1730. While compare to the explicit solution (23) the computation of ( ) , k k m n from the recurrence relations (22) is very simple, the following interesting relation follows from (23) by direct substitution Hence the difference between two consecutive square triangular numbers is the square root of another square triangular number. Now we note that the system (19) can be written as and, when n is odd, and hence the right side is a perfect square for ( ) 2 1 k n p n = + . Therefore, the product of ( ) and hence the right side is a perfect square for 2 k n pn = (which is always even). Therefore, two times the product of 2 p consecutive triangular numbers is a perfect square for each 1 p ≥ and 1 k ≥ . In particular, for 2 p k = = ,  it follows that if the triangular number n t is square, then ( ) 4 1 n n t + is also square.
Since 1 t is square, it follows that there are infinite number of square triangular numbers. This clever observation was reported in 1662, see Pietenpol et al. [18].
From this, the first four square triangular numbers, we get are 1 8 288 , , t t t and 332928 t .  There are infinitely many triangular numbers that are simultaneously expressible as the sum of two cubes and the difference of two cubes. For this, Burton [19] begins with the identity and observed that if k is odd then this equality can be written as which is the same as  x For this, first we note that integers , 2 1 t t + and 1, 2 1 t t + + are coprime, i.e., they do not have any common factor except 1. We also recall that if the product of coprime numbers is a p-th power, then both are also of p-the power. Now let n be even, i.e.,

Square Numbers S n
In this arrangement rows as well as columns contain 1, 2, 3, 4, , n  dots, (see Figure 5).
From Figure 5 it is clear that a square made up of ( )  i.e., the differences between successive nested squares produce the sequence of odd numbers. From (30) it follows that   Figure 6 provides proof of (31) without words. Here odd integers, one block, three blocks, five blocks, and so on, arranged in a special way. We begin with a single block in the lower left corner; three shaded blocks surrounded it to form a 2 2 × square; five unshaded blocks surround these to form a 3 3 × square; with the next seven shaded blocks we have a 4 4 × square; and so on. The diagram makes clear that the sum of consecutive odd integers will always yield a (geometric) square. Comparing Figure 1 and Figure 5 or Figure 2 and Figure 6, it is clear that n-th square number is equal to the n-th triangular number increased by its predecessor, i.e., The following equality is of exceptional merit ( ) ( ) Therefore, ( )( ) is the generating function of all square numbers.
From (38) it also follows that the generating function for all gnomonic numbers is ( ) ( ) ( )  The sum of the first n square numbers is given in (11). For the exact sum of the reciprocals of the first n square numbers no formula exists; however, the problem of summing the reciprocals of all square numbers has a long history and in the literature it is known as the Basel problem. Euler in 1748 considered sin , 0 x x x ≠ which has roots at , 1 n n ± π ≥ . Then, he wrote this function in terms of infinite product The above demonstration of Euler is based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof. Now in the literature for (39) several different proofs are known, e.g., for a recent elementary, but clever demonstration, see Murty [23].
 The following result provides a characterization of all Pythagorean triples, i.e., solutions of (18): Let u and v be any two positive integers, with u v > , then the three numbers form a Pythagorean triple. If in addition u and v are of opposite parity-one even and the other odd-and they are coprime, i.e., that they do not have any common factor other than 1, then ( ) , , a b c is a primitive Pythagorean triple. The converse, i.e., any Pythagorean triple is necessarily of the form (40) also holds. For the proof and history of this result see, Agarwal [14]. From (18), (32), and (40) the following relations hold The relation (30) can be written as ( ) ( )    (18) and (32)

Rectangular (Oblong, Pronic, Heteromecic) Numbers R n
In this arrangement rows contain ( )  i.e., we add successive even numbers, or two times triangular numbers. It also follows that rectangular number 1 n R + is made from n R by adding an L--shaped border (a gnomon), with ( ) i.e., the differences between successive nested rectangular numbers produce the sequence of even numbers. Thus the odd numbers generate a limited number of forms, namely, squares, while the even ones generate a multiplicity of rectangles which are not similar. From this the Pythagoreans deduced the following correspondence: odd limited and even unlimited.

↔ ↔
We also have the relations   (31) and (46) it follows that (44) shows that the product of two consecutive positive integers n and ( ) 1 n + is the same as two times n-th triangular numbers. According to historians with this relation Pythagoreans' enthusiasm was endless. Relation (45) reveals that every even integer 2n is the difference of two consecutive rectangular numbers n R and  There is no rectangular number which is also a perfect square, in fact, the equation ( ) 2 1 n n m + = has no solutions (the product of two consecutive integers cannot be a prefect square).
 To find all rectangular numbers which are also triangular numbers, we need to find integer solutions of the equation ( ) ( )  Fibonacci numbers denoted as n F are defined by the recurrence relation or the closed from expression  Lucas numbers which are also triangular are 1, 3, 5778, i.e., 1 2 107 , , t t t . From the above explicit expressions the following relations can be obtained easily

Pentagonal Numbers P n
The pentagonal numbers are defined by the sequence 1, 5,12, 22, 35, 51, , i.e., beginning with 5 each number is formed from the previous one in the sequence by adding the next number in the related sequence   Comparing (50) with (3), we have 1, and hence from (4) it follows that It is interesting to note that n P is the sum of n integers starting from n, i.e.,  Relation (51) shows that pentagonal number n P is the one-third of the ( ) 3 1 n − -th triangular number, whereas relation (53) reveals that it is the sum of n-th triangular number and two times of ( ) is the generating function of all pentagonal numbers.
 From (2), (11) and (51) it is easy to find the sum of the first n pentagonal numbers ( )  To find all square pentagonal numbers, we need to find integer solutions of the equation ( )  To find all pentagonal numbers which are also triangular numbers, we need to find integer solutions of the equation ( ) ( ) . This equation can be written as Pell's equation 2
Thus, n-th hexagonal number is defined as is the generating function of all hexagonal numbers.
 From (2), (11), and (60) it is easy to find the sum of the first n hexagonal numbers ( )( ) here, n a and n b are as in (25).  To find all hexagonal numbers which are also rectangular numbers, we need to find integer solutions of the equation ( ) ( )  To find all hexagonal numbers which are also pentagonal numbers, we need to find integer solutions of the equation ( ) ( )

Generalized Pentagonal Numbers (Centered Hexagonal Numbers, Hex Numbers) (GP) n
The generalized pentagonal numbers are defined by the sequence 1, 7,19, 37, 61, , i.e., beginning with 7 each number is formed from the previous one in the sequence by adding the next number in the related sequence = + + + = + , and so on (see Figure 11). These numbers are also called centered hexagonal numbers as these represent hexagons with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. These numbers have practical applications in materials logistics management, for example, in packing round items into larger round containers, such as Vienna sausages into round cans, or combining individual wire strands into a cable.
Thus, n-th generalized pentagonal number is defined as Hence, from (2) it follows that 3 0 is the generating function of all generalized pentagonal numbers.   (15) and (67) it is easy to find the sum of the first n generalized pentagonal numbers ( ) ( ) ( )  To find all square generalized pentagonal numbers, we need to find integer solutions of the equation  To find all generalized pentagonal numbers which are also triangular numbers, we need to find integer solutions of the equation . This equation can be written as Pell's equation and 2 1 a n = − . For this, corresponding to (22) the system is  There is no generalized pentagonal number which is also a rectangular number, in fact, the equation  To find all generalized pentagonal numbers which are also pentagonal numbers, we need to find integer solutions of the equation . This equation can also be written as Pell's equation 2 − and 2 1 a n = − . For this, corresponding to (22) the system is  To find all generalized pentagonal numbers which are also hexagonal numbers, we need to find integer solutions of the equation This equation can also be written as Pell's equation and 2 1 a n = − . For this, corresponding to (22) the system is  = + + + = + , and so on (see Figure 12).
Thus, n -th heptagonal number is defined as    To find all heptagonal numbers which are also pentagonal numbers, we need to find integer solutions of the equation ( ) ( ) = + + + = + , and so on (see Figure 13).
Thus, n-th octagonal number is defined as  ,     To find all octagonal numbers which are also heptagonal numbers, we need to find integer solutions of the equation ( ) ( )  1.2433209262; is the digamma function defined as the logarithmic derivative of the gamma function ( )    and so on (see Figure 15).
Hence, n-th decagonal number is defined as   , and so on (see Figure 16).
Hence, n-th tetrakaidecagonal number is defined as     To find all tetrakaidecagonal numbers which are also triangular numbers, we need to find integer solutions of the equation ( ) ( )  To find all tetrakaidecagonal numbers which are also hexagonal numbers, we need to find integer solutions of the equation ( ) ( )    = + + + = + , and so on (see Figure 19).
Hence, n-th centered pentagonal number is defined as   , and so on (see Figure 25).
Hence, n-th centered tetrakaidecagonal number is defined as In the literature often the above representation is referred to as Pascal's triangle. Now noting that numbers in each row are odd, so the general term in view of (31) can be written as Taking successively 1, 2, 3, , k n =  in the above relation, adding these n equations, and observing that ( ) ( ) ( ) is the count of number of points in a body-centered cubic pattern within a cube that has 1 n + points along each of its edges. First few centered cubic numbers are 1,9,35,91,189,341. Clearly, no centered cubic number is prime. Further, the only centered cube number that is also a square number is 9. The generating function for all centered cube numbers is ( )

Tetrahedral Numbers (Triangular Pyramidal Numbers) T n
These numbers count the number of dots in pyramids built up of triangular numbers. If the base is the triangle of side n, then the pyramid is formed by placing similarly situated triangles upon it, each of which has one less in its sides than that which precedes it (see Figure 27).
In general, the nth tetrahedral number T n is given in terms of the sum of the first n triangular numbers, i.e.,    To find the sum of the reciprocals of all tetrahedral numbers we follow as in (16) and (17), to obtain ( )( )  As in (173)  The numbers ( )    These numbers represent the number of spheres in an octahedral formed from close-packed spheres. Descartes initiated the study of octahedral numbers around 1630. In 1850, Pollock conjectured that every positive integer is the sum of at most 7 octahedral numbers, which for finitely many numbers have been proved by Brady [33]. The difference between two consecutive octahedral numbers is a centered square number, i.e., In (189), the function Φ is the Dirichlet beta function. Thus, generalized pentagonal pyramidal numbers are the same as cubic numbers.

Heptagonal Pyramidal Numbers (HEPP) n
These numbers count the number of dots in pyramids built up of heptagonal numbers. First few heptagonal pyramidal numbers are 1,8,26,60,115,196,308,456. In general, the nth heptagonal pyramidal number ( ) n HEPP is given in terms of the sum of the first n heptagonal numbers, i.e.,

Stella Octangula Numbers (SO) n
The word octangula for eight-pointed star was given by Johannes Kepler

Biquadratic Numbers (BC) n
A biquadratic number can be written as a product of four equal factors of natural numbers. Thus, 1, 16, 81, 256, 625, 1296 These numbers can be represented as regular discrete geometric patterns, see Deza [3]. In biochemistry, the pentatope numbers represent the number of possible arrangements of n different polypeptide subunits in a tetrameric (tetrahedral) protein.
 Two of every three pentatope numbers are also pentagonal numbers. In fact, the following relations hold ( ) ( ) Singapore, for his generous guidance and encouragement in the preparation of this article. He is also thankful to Professor Chao Wang and referees for their comments.

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