Elementary Number Theory with Programming
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"It consists of nine chapters, all including the corresponding programs along with their mathematical content. The mathematical structure is also interesting and well-formed starting from special numbers, primes and Pell equation, to Pascal's triangle, prime decomposition and modular arithmetic and finishing with number-theoretic functions, the Euler Phi-function, sums and partitions and the classical application to cryptography. It is also remarkable that the main scope of the programs is defined before their use from the reader, providing him the best orientation for his study." (Zentralblatt MATH 2016)More details
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1
SPECIAL NUMBERS: TRIANGULAR, OBLONG, PERFECT, DEFICIENT, AND ABUNDANT
We start our introduction to number theory with definitions, properties, and relationships of several categories of numbers.
TRIANGULAR NUMBERS
Triangular numbers are those that can be written as the sum of a consecutive series of (whole) numbers beginning with 1. Thus 6 is triangular because it is the sum of the first three numbers: 6 = 1 + 2 + 3. The first few triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, and 55. We denote the nth triangular number by tn. Thus t5 = 1 + 2 + 3 + 4 + 5 = 15. More generally,
(1.1)Our first program, calculating a specific triangular number, shows the format of an HTML document. The first line specifies the doctype. The rest is an html element, starting with <html> and ending with </html>. Within the html element is a head element and a body element. In this case, the body element is empty. The head element contains a meta tag specifying the character type (it can be omitted), a title, and a script element. All the action is in the script element.
The code makes use of standard programming constructs such as variables and functions and for-loops (if you don't understand what these terms are, please consult any beginner book on programming. Shameless plug: go to The Essential Guide to HTML5: Using Games to Learn HTML5 and JavaScript, http://www.apress.com/9781430233831).
The specific triangular number we want is specified in the coding by setting the variable n . This is termed hard-coding. The computation is done using a for-loop. The for-loop adds up the values from 1 to n , exactly following Equation 1.1. The built-in method document.write writes out the result.
The challenge in Exercise 1 is to compare coding using Equation 1.1 versus Equation 1.2. The challenge is that computers are very fast. I use the built-in Date function with the method getTime to get the number of milliseconds from a base date at the start and after the computation. It turns out that computing the millionth triangular number takes 3 ms! You can experiment with different values. Using the formula given in Equation 1.2 would be much, much faster. Give it a try.
The nth triangular number is given by the formula:
(1.2)Example:
Example:
Write 6 + 7 + 8 + 9 + 10 + 11 as the difference of two triangular numbers. We observe that 6 + 7 + 8 + 9 + 10 + 11 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11) - (1 + 2 + 3 + 4 + 5), which is t11 - t5.
Example:
Generalize the previous example to any consecutive sum such as 45 + 46 + + 987. Note that a + (a+1) + (a+2) +.+ b = (1 + 2 + 3 + . + b)-(1 + 2 + 3 + . + (a-1)) = tb-ta-1. By letting a = 6 and b = 11, we get the result of the previous example.
It should be noted that
(1.3)The sum of any two consecutive triangular numbers is a square. For example, t4 + t3 = 10 + 6 = 16 = 42 and t5 + t4 = 15 + 10 = 25 = 52. This is expressed by the formula
(1.4)Example:
Verify (1.4) for n = 10. We have t10 + t9 = 55 + 45 = 102.
Example:
Find two triangular numbers whose sum is 900. Since 900 = 302, we have n = 30. Then using (1.4), .
The sum of the reciprocals of all the triangular numbers is 2. Formally,
(1.5)OBLONG NUMBERS AND SQUARES
A positive integer of the form n(n + 1) is called oblong. The nth oblong number is the sum of the first n even numbers. To see this, observe that the nth even number is 2n. Then we have , the nth oblong number. What about the sum of the first n odd numbers? The nth odd number is 2n - 1. So , in which -1 appears n times. We then get . So the sum of the first n odd numbers is n2.
Example:
The sum of the first 5 odd numbers is 25. (Check this: 1 + 3 + 5 + 7 + 9 = 25.) More impressively, the sum of the first 100 odd numbers is 1002 = 10,000.
The great French mathematician LaGrange (1736-1813) showed in the late eighteenth century that every positive number can be written as a sum of four or fewer squares. Thus, for example, 30 = 25 + 4 + 1.
Number theorists are fond of numbers, such as 40, which are the sum of only two squares (e.g., 40 = 36 + 4).
The Pythagoreans computed the sum of the first n powers of 2. Let
- . Then
- .
Now subtract Equation (a) from Equation (b), and we get S = 2n - 1. We have, then, the following formula:
(1.6)With a minor change in the proof of (1.6), we obtain an analogous formula for the sum of the first n powers of any base. Let . Then . Subtract the first equation from the second, and we get (a - 1)S = an - 1. Upon division by a - 1, we obtain the following formula:
(1.7)DEFICIENT, ABUNDANT, AND PERFECT NUMBERS
The Pythagoreans classified all numbers as deficient, abundant, or perfect. Given a number, find all of its proper factors, that is, all numbers that go into it (with the exclusion of the given number). The proper factors of 30, for example, are 1, 2, 3, 5, 6, 10, and 15.
Generating a list of the factors of a number is easy in JavaScript (and other programming languages), though it appears tedious to us. The modulo operation, %, determines the remainder. So if n is the number and f is a candidate factor, then
n % f will produce the remainder of n divided by f. If this is 0, then f is a factor. If f < n, then f is a proper factor.
The program uses a for-loop going from 1 up to but not including n. If it is a factor, the number is written out in the html document using document.write and a variable count is incremented.
If the sum of the proper factors of n is less than n, we call n deficient. If the sum exceeds n, it is called abundant. If the sum equals n, we call it perfect. For example, 8 is deficient since 1 + 2 + 4 < 8, 18 is abundant since 1 + 2 + 3 + 6 + 9 > 18, and 28 is perfect since 1 + 2 + 4 + 7 + 14 = 28. The smallest perfect number is 6. The first few perfect numbers are 6, 28, 496, and 8128. It is not known today whether there are infinitely many perfect numbers. Moreover, all known perfect numbers are even. No one knows if there are any odd perfect numbers! Incidentally, the smallest abundant odd number is 945, while the smallest abundant even number is 12.
The program to characterize a number as deficient, perfect, or abundant was made by modifying the previous one that listed and counted the number of proper factors. To make the determination of whether a number n is deficient, perfect, or abundant, the program has to add up the proper factors. So the statement count++ is removed and the statement
sum += i; is inserted. By the way, this is shorthand for taking the original value of the variable sum, adding one to it and then assigning that back to the variable sum.
sum = sum + i; I also changed the name of the function to addUpFactors. I tested the program using the specific numbers given in the text.
The Pythagoreans found an amazing method for finding perfect numbers. They observed, using (1.6), that sums of the form are prime for certain values of n and are...
