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 t_n. Thus t₅ = 1 + 2 + 3 + 4 + 5 = 15. More generally,

(1.1)

The nth triangular number is given by the formula:

(1.2)

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)

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.

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

1. . Then
2. .

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.

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 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 composite for others. (A number is prime if its only factors are 1 and itself. 7, 19, and 31 are examples of primes. A composite number has proper factors other than 1. Thus 20 is composite.) The following sums, for example, are prime:

In each of these equations, multiply the greatest number on the left by the number on the right, yielding 2 × 3 = 6, 4 × 7 = 28, 16 × 31 = 496, and 64 × 127 = 8128. These products, 6, 28, 496, and 8128 are perfect. Whenever the sum of the first n powers of 2 is prime, this procedure yields a perfect number! Using (1.6), the sum of the first n powers of 2 is 2ⁿ − 1, so the perfect number is of the form 2ⁿ⁻¹(2ⁿ − 1). The prime sum, 2ⁿ − 1, is then called a Mersenne prime, in honor of the eighteenth century French mathematician. It was shown in the eighteenth century by the great Swiss mathematician Leonhard Euler (1707–1783) that all even perfect numbers are of the form 2ⁿ⁻¹(2ⁿ − 1).

A conjecture is that no odd number (odd number >1) is perfect. One of the exercises and one of our programs tests this conjecture on the first 1000 odd numbers.

The Pythagoreans believed that if two friends wore amulets, one with 220 and the other with 284, they would fortify their friendship. This is because the sum of the proper factors of either one of these numbers equals the other number, that is,

We call a pair of numbers with this property, amicable numbers. 1184 and 1210 comprise the next pair of amicable numbers, since

EXERCISES

An asterisk (*) indicates that the exercise can be developed into a research project.

1. Write a program to find the nth triangular number, t_n, using formula (1.1). Then write a program using (1.2). Compare the two procedures for very large values of n. A program for the first part of this is included at the end of the chapter.
2. Write a program to find out whether a given number is a square.
3. Find, using a partial fraction decomposition, the sum of the reciprocals of the first n triangular numbers, that is, find . Then write a program to do this.
4. ^*The ancient Egyptians expressed every proper fraction except as the sum of fractions with 1s in the numerator. Thus, is equivalent to is equivalent to as .
   1. Verify the identity .
   2. Using the strategy of starting off by writing a fraction a/b as the sum of 1/b + 1/b + … (a times) and using the identity verified in part a repeatedly until all fractions are distinct, write a program to express every fraction as the sum of fractions with distinct denominators and numerators equal to 1.
5. ^*Note that the sum and difference of the triangular numbers 15 and 21 are triangular. Verify that this is also the case for the triangular numbers 780 and 990. Find 100 more cases.
6. Find 100 triangular numbers that are squares.
7. From the first 1000 triangular number, find ones that are the sum of two other triangular numbers.
8. ^*Find 100 oblong numbers that are products of an oblong number and a square.
9. Show that after 3, the next 100 triangular numbers are composite (not prime). Then prove this for all triangular numbers after 3.
10. Write each of the numbers from 1 to 1000 as the sum of three or fewer triangular numbers.
11. ^*Write each of the numbers from 1 to 1000 as the sum of four or fewer squares. For which of these numbers can this be done in more than one way? For example, 50 = 49 + 1 = 25 + 25 = 36 + 9 + 4 + 1 = 16 + 16 + 9 + 9.
12. ^*What proportion of the first 1000 numbers can be written using two or fewer squares?
13. Write a program that lists the proper divisors of a given number. A program for this is given at the end of the chapter.
14. Write a program to find the sum of the proper divisors of a given number.
15. Modify the program of the previous exercise to decide whether a given number is deficient, perfect, or abundant. A program for this is included at the end of the chapter. You can improve this in various ways, including having the user enter the number. Look ahead to an example in Chapter 2 that shows how to get user input.
16. ^*Write a program to check for perfect numbers within a range. You can set the endpoints of the range within the program. You can research to find a list of the known perfect numbers AND to determine the biggest integer value that can be represented in regular JavaScript.
17. ^*A number is called semi-perfect if it is the sum of some (but not all) of its proper divisors. 12 is the smallest semi-perfect number since 12 = 6 + 4 + 2. Find the next 50 semi-perfect numbers.
18. ^*If a given number is abundant, determine if it is semi-perfect.
19. Show that 2ⁿ is deficient for all n ≤ 25. Then show it is deficient for all n.
20. ^*Verify that 945 is the smallest odd abundant number. Find the next 10 odd abundant numbers. Do they seem to be getting further apart?
21. Observe that the square of the triangular number 6 is also triangular. Verify that this does not occur for any other triangular number (except 1) up to t₁₀₀₀.
22. It has been conjectured that no odd number is perfect. Verify this for the first 1000 odd numbers. See last example. You can decide on other ways to present the findings and also change the limit.

<html>
<head>
<title>Triangular Numbers</title>
<script>
var n = 1000000;
var start = new Date();
start = start.getTime();

function init(){
   var sum = 0;
 for(i=1;i<=n;i++){
   sum+=i;
 }
 now = new Date();
 now = now.getTime();
 elapsed = (now - start);
 document.write("The "+n+
     "th triangular number is "+sum+".<br/>");
 document.write("Elapsed time was "+elapsed+"milliseconds.");
}
init();
</script>
</head>
<body>
</body>
</html>

Proper Factors

<!DOCTYPE HTML>
<html>
<head>
<title>Proper factors</title>
<script>
var n = 30;

var count = 0;
function countUpFactors(n){
  document.write("Proper factors of "+n+" are:
  <br/>");
  for (var i=1;i<n;i++){
      if ((n%i)==0){
         count++;
         document.write(i+"<br/>");
      }
  }
  document.write("The number of proper factors
  of "+n+" is "+count+".");  
}
countUpFactors(n);
</script>
</head>
<body>
</body>
</html>

Classifying Number as Deficient, Perfect, or Abundant

<!DOCTYPE HTML>
<html>
<head>
<title>Perfect or </title>
<script>
function addUpFactors(n){
  document.write("Proper factors of "+n+" are:
  <br/>");
  var sum = 0;
  for (var i=1;i<n;i++){
    if ((n%i)==0){
       document.write(i+"<br/>");
       sum += i;
    }
 }
 
 document.write("The sum of the proper factors
 of "+n+" is "+sum+", so "+n+" is ");
 if (sum<n) {
    document.write("deficient.");
 }
 else if (sum==n) {
    document.write("perfect.");
 }
 else {
    document.write("abundant.");
 }
 document.write("<br/>");
}
addUpFactors(6);
addUpFactors(8);
addUpFactors(18);
addUpFactors(28);
addUpFactors(945);
addUpFactors(8128);
</script>
</head>
<body>
</body>
</html>

Checking the Conjecture on No Perfect Odd Number (up to 1000)

<!DOCTYPE HTML>
<html>
<head>
<title>Sort Odd numbers </title>
<script>
function classify(n){

  var sum = 0;
  for (var i=1;i<n;i++){
     if ((n%i)==0){
        //document.write(i+"<br/>");
        sum += i;
     }
  }

  if (sum<n) {
     return -1;
  }
  else if (sum==n) {
     return 0;
  }
  else {
     return 1;
  }
  document.write("<br/>");
}
 function sortOdds(limit) {
 var perfect = 0;
 var deficient = 0;
 var abundant = 0;
 for (var j=1;j<=limit;j++) {
    var n=2*j+1;
    c = classify(n);
    if (c==-1){
     deficient++;
    }
    else if (c==1) {
     abundant++;
    }
    else {
     perfect++;
    }
}
  document.write("First "+limit+" odd numbers:
  <br/>");
  document.write ("deficient: "+deficient+
  "<br/>");
  document.write("abundant: "+abundant+
  "<br/>");
  document.write("perfect: "+perfect+"<br/>");
 }
 sortOdds(1000);
</script>
</head>
<body>
</body>
</html>