The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX.
Both methods appear in documents from the Roman era, even within the same document, double subtractives occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases.
There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD.
Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary and octal numbers can be specified similarly
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers.
A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, two classes of such formal definitions were constructed, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Highly composite number
A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan, the related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The initial or smallest 38 highly composite numbers are listed in the table below, the number of divisors is given in the column labeled d. The table below shows all the divisors of one of these numbers, the 15, 000th highly composite number can be found on Achim Flammenkamps website. Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, except in two special cases n =4 and n =36, the last exponent ck must equal 1. It means that 1,4, and 36 are the only square highly composite numbers, saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials. Note, that although the above described conditions are necessary, they are not sufficient for a number to be highly composite.
For example,96 =25 ×3 satisfies the conditions and has 12 divisors but is not highly composite since there is a smaller number 60 which has the same number of divisors. If Q denotes the number of composite numbers less than or equal to x. The first part of the inequality was proved by Paul Erdős in 1944 and we have 1.13862 < lim inf log Q log log x ≤1.44 and lim sup log Q log log x ≤1.71. Highly composite numbers higher than 6 are abundant numbers, one need only look at the three or four highest divisors of a particular highly composite number to ascertain this fact. It is false that all composite numbers are Harshad numbers in base 10. The first HCN that is not a Harshad number is 245,044,800, which has a sum of 27. 10 of the first 38 highly composite numbers are highly composite numbers. The sequence of composite numbers is a subset of the sequence of smallest numbers k with exactly n divisors. A positive integer n is a composite number if d ≥ d for all m ≤ n. The counting function QL of largely composite numbers satisfies c ≤ log Q L ≤ d for positive c, d with 0.2 ≤ c ≤ d ≤0.5.
Because the prime factorization of a composite number uses all of the first k primes
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix.
There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors.
It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can be thought of as real numbers, and if they are, x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n. The study of these dates back to Aristotle. They are called oblong numbers, heteromecic numbers, or rectangular numbers, the rectangular number name has been applied to the composite numbers. The first few numbers are,0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,462 …. The nth pronic number is the difference between the odd square 2 and the st centered hexagonal number. The sum of the reciprocals of the numbers is a telescoping series that sums to 1,1 =12 +16 +112 ⋯ = ∑ i =1 ∞1 i. The partial sum of the first n terms in this series is ∑ i =1 n 1 i = n n +1, the nth pronic number is the sum of the first n even integers. It follows that all numbers are even, and that 2 is the only prime pronic number. It is the only number in the Fibonacci sequence. The number of entries in a square matrix is always a pronic number. The fact that consecutive integers are coprime and that a number is the product of two consecutive integers leads to a number of properties.
Each distinct prime factor of a number is present in only one of the factors n or n+1. Thus a pronic number is squarefree if and only if n and n +1 are squarefree, the number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n +1. If 25 is appended to the representation of any pronic number. This is because 2 =100 n 2 +100 n +25 =100 n +25
Virginia is a state located in the Mid-Atlantic region of the United States, as well as in the historic Southeast. The geography and climate of the Commonwealth are shaped by the Blue Ridge Mountains and the Chesapeake Bay, the capital of the Commonwealth is Richmond, Virginia Beach is the most populous city, and Fairfax County is the most populous political subdivision. The Commonwealths estimated population as of 2014 is over 8.3 million, the areas history begins with several indigenous groups, including the Powhatan. In 1607 the London Company established the Colony of Virginia as the first permanent New World English colony, slave labor and the land acquired from displaced Native American tribes each played a significant role in the colonys early politics and plantation economy. Although the Commonwealth was under one-party rule for nearly a century following Reconstruction, the Virginia General Assembly is the oldest continuous law-making body in the New World. The state government was ranked most effective by the Pew Center on the States in both 2005 and 2008 and it is unique in how it treats cities and counties equally, manages local roads, and prohibits its governors from serving consecutive terms.
Virginias economy changed from agricultural to industrial during the 1960s and 1970s. Virginia has an area of 42,774.2 square miles, including 3,180.13 square miles of water. Virginias boundary with Maryland and Washington, D. C. extends to the mark of the south shore of the Potomac River. The southern border is defined as the 36° 30′ parallel north, the border with Tennessee was not settled until 1893, when their dispute was brought to the U. S. Supreme Court. The Chesapeake Bay separates the portion of the Commonwealth from the two-county peninsula of Virginias Eastern Shore. The bay was formed from the river valleys of the Susquehanna River. Many of Virginias rivers flow into the Chesapeake Bay, including the Potomac, Rappahannock and James, the Tidewater is a coastal plain between the Atlantic coast and the fall line. It includes the Eastern Shore and major estuaries of Chesapeake Bay, the Piedmont is a series of sedimentary and igneous rock-based foothills east of the mountains which were formed in the Mesozoic era.
The region, known for its clay soil, includes the Southwest Mountains around Charlottesville. The Blue Ridge Mountains are a province of the Appalachian Mountains with the highest points in the state. The Ridge and Valley region is west of the mountains and includes the Great Appalachian Valley, the region is carbonate rock based and includes Massanutten Mountain. The Cumberland Plateau and the Cumberland Mountains are in the southwest corner of Virginia, in this region, rivers flow northwest, with a dendritic drainage system, into the Ohio River basin
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front.
For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients.
Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
The Padovan sequence is the sequence of integers P defined by the initial values P = P = P =1, and the recurrence relation P = P + P. The first few values of P are 1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265. The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom, Hans van der Laan, Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996 and he writes about it in one of his books, Math Hysteria, Fun Games With Mathematics. The above definition is the one given by Ian Stewart and by MathWorld, other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets. This is a property of recurrence relations, the Perrin sequence can be obtained from the Padovan sequence by the following formula, P e r r i n = P + P. e. The Padovan sequence satisfies the identity P2 − P P = P.
The Padovan sequence is related to sums of binomial coefficients by the following identity, P = ∑2 m + n = k = ∑ m = ⌈ k /3 ⌉ ⌊ k /2 ⌋. For example, for k =12, the values for the pair with 2m + n =12 which give non-zero binomial coefficients are, and, + + =1 +10 +1 =12 = P. The Padovan sequence numbers can be written in terms of powers of the roots of the equation x 3 − x −1 =0 and this equation has 3 roots, one real root p and two complex conjugate roots q and r. Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r, P = a p n + b q n + c r n where a, b and c are constants. Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n, and P − a p n tends to zero. For all n ≥0, P is the integer closest to p n −1 s, the ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the ratio does to the Fibonacci sequence. P is the number of ways of writing n +2 as a sum in which each term is either 2 or 3.
This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as, ∑ n =0 ∞ P α n = α2 α3 − α −1. A Padovan prime is P that is prime, the first few Padovan primes are 2,3,5,7,37,151,3329,23833. Also, if you count the number of As, Bs and Cs in each string, for the nth string, you have P As, P Bs, the count of BB pairs, AA pairs and CC pairs are Padovan numbers
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China.
The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary.
Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U.
S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century.
The number of pounds in an American short hundredweight, in Greece, India and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
Square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers solve the problem of counting the number of squares in an n × n grid. The first few square pyramidal numbers are,1,5,14,30,55,91,140,204,285,385,506,650,819 and this is a special case of Faulhabers formula, and may be proved by a mathematical induction. An equivalent formula is given in Fibonaccis Liber Abaci, in modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial L of a polyhedron P is a polynomial that counts the number of points in a copy of P that is expanded by multiplying all its coordinates by the number t. The Ehrhart polynomial of a pyramid base is a unit square with integer coordinates. The square pyramidal numbers can be expressed as sums of binomial coefficients, the smaller tetrahedral number represents 1 +3 +6 + ⋯ + T and the larger 1 +3 +6 + ⋯ + T.
Offsetting the larger and adding, we arrive at 1,1 +3,3 +6 …, Square pyramidal numbers are related to tetrahedral numbers in a different way, P n =14. The sum of two square pyramidal numbers is an octahedral number. Augmenting a pyramid whose base edge has n balls by adding to one of its faces a tetrahedron whose base edge has n −1 balls produces a triangular prism. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism and this geometric dissection leads to another relation, P n = n −. Besides 1, there is one other number that is both a square and a pyramid number,4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918, in the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes. Also, P n = − which is the difference of two pentatope numbers and this can be seen by expanding, n − n = n = n and dividing through by 24. A common mathematical puzzle involves finding the number of squares in a n by n square grid.
This number can be derived as follows, The number of 1 ×1 boxes found in the grid is n2, the number of 2 ×2 boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of 2 ×2 boxes, the number of k × k boxes found in the grid is 2. These can be counted by counting all of the possible upper-left corners of k × k boxes and it follows that the number of squares in an n × n square grid is, n 2 +2 +2 +2 + … +12 = n 6