Topics in Number Theory
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Following are some topics in number theory.

Contents of this Page
Pascal's Triangle
Perfect Numbers
Fermat's Last Theorem
Magic Squares
Moessner's Magic

Pascal's Triangle


Pascal's triangle was first introduced by the Chinese mathematician Yang Hui, but it got it's name from Blaise Pascal who 500 years later rediscovered it along with Omar Khayyam.

The triangle is used to look for the probability of any particular event to occur. There are many other things that can be found in the triangle. Listed below are a few of them and how to achieve them.

How to make Pascal's Triangle. Row 0 is the first row, it will have a 1. Row 1 is actually the second row it will have 1 and 1, but not to be confused with 11. The next row is the numbers 1 and 2 and 1. Now how did we get these numbers? 1 is ALWAYS going to be the first number in the row, but in order to make the triangle grow you add the two numbers above. Example: 1 + 2 = 3 and 2 + 1 = 3, so for the next line we will have 1 (always on the outside) and 3 and 3 and then 1 again. The next line gets even bigger, 1 (outside again) 1 + 3 = 4, and 3 + 3 = 6, and 3 + 1 = 4, and then that 1 again.

This can go on as long as anyone wants it to go.

The first 5 powers of 11 are in the top of the triangle. 110 = 1, 111 = 11, 112 = 121, 113 = 1331, and 114 = 14641. When these numbers are stacked in a pyramid it will form the top part of the triangle.

If you start at any given 1 and go diagonally down and then make a one step left you will find the answer to the numbers that you just followed. For example: 1 + 2 + 3 + 4 + 5 = 15, or try any other number 1 + 8 + 36 + 120 + 330 = 495.

Take any number next to one and make a triangle with the number directly beside it and the one below the two, when this number is squared you will have the answer of the two additional numbers when they are added together. For example: 52 = 25, looking to the right of 5 is 10, now look at the number below the 5 and 10, should be 15, now 10 + 15 = 25.

Any row starting with a 1 can be added straight across to find the sum of 2 to the power of that rows number. Example: 20 = 1, 21 = the addition of the next row (1 + 1) = 2, 22 = (1 + 2 + 1) = 4, 23 = (1 + 3 + 3 + 1) = 8, what about the 10th row? 1 + 9 + 36 + 84 + 126 + 126 + 84 + 36 + 9 + 1 = 512, which is 29. WOW!! Is this fun....

The sum of the numbers in a diagonal line downward are the first numbers of the Fibonnacci Sequence. Example: 1 is alone so = 1, then the next would also be 1 = to 1, then 1 and 1 = 2, 2 and 1 = 3, 1 and 3 and 1 = 5, 3 and 4 and 1 = 8, 1 and 6 and 5 and 1 = 13, this can be done all the way down the triangle (as the triangle gets bigger it gets a little confusing, so make sure you color out the numbers you have already used). Now take the = numbers like this: 1 + 1= the next number 2. 1 + 2 = the next number 3, 2+3 = the next number 5, 3 + 5 = the next number 8, so forth and so on. By the way Fibonacci's sequence were used to discribe a curve found in many string instruments.

Contributed by Cris Edelman


  1. Pascal's Triangle
  2. All You Ever Wanted to Know About Pascal's Triangle and more.
  3. Pascal's Traingle

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Perfect Numbers


The Pythagoreans produced a theory of numbers comprised of numerology and scientific speculation. In their numerology, even numbers were feminine and odd numbers masculine. The numbers also represented abstract concepts such as 1 stood for reason, 2 stood for opinion, 3 stood for harmony, 4 stood for justice, and so on. Their arithmetica had a theory of special classes of numbers. There were “perfect” numbers of two kinds. The first kind included only 10, which was basic to the decimal system and the sum of the first four numbers 1 + 2 + 3 + 4 = 10. The second kind of “perfect” numbers were those equal to the sum of their proper divisors.

A perfect number is a positive integer that is equal to the sum of it divisors. However, for the case of a perfect number, the number itself is not included in the sum. The Greeks called a number such as 6 or 28 a “perfect” number because the sum of the proper divisors in each case is equal to the number; the proper divisors of 6 are 1, 2, and 3, and their sum is 6.

Although perfect numbers are regarded as arithmetical curiosities, their study has helped to develop the theory of numbers. Euclid proved that a number n of the form (2n-1)*2n-1 is a perfect number if the factor 2n-1 is prime. For example, if n assumes the value 2, 3, 5, or 7, the expression 2n-1 takes on the value 3, 7, 31, or 127, all of which are prime. For these values of n we obtain the perfect numbers 6, 28, 496, and 8,128.

The Neoplatonists Nicomachus of Gerasa and Iamblichus of Chalcis listed these perfect numbers and concluded that they follow a pattern: They alternately end in a 6 or an 8, and there is one perfect number for each interval from 1 to 10, 10 to 100, 100 to 1,000 and 1,000 to 10,000. They conjectured that both parts of the pattern would continue, but in this they were wrong. The fifth perfect number, which was discovered in the fifth century, corresponds to n = 13 and is 33,550,336, with eight digits rather than six. In addition, the sixth perfect number, like the fifth, ends with a six.

In 1961, the twentieth perfect number was found. It contains 2,663 digits in the decimal representation and corresponds to the case where n = 4,423. Today, twenty-three perfect numbers are known. The largest of these is 211,212(211,213-1), which contains 6,751 digits. It is not known whether there are an infinite number of perfect numbers.

In 1757, Leonhard Euler probed that every even perfect number must be of Euclid’s form. It has also been proven that every even perfect number must end in six or eight and if it ends in six, the digit preceding it must be odd. No one has as yet discovered an odd perfect number, but it is known that none exist below 1020.

Contributed by Kristen Shelton


Calinger, Ronald, Classics of Mathematics, Prentice Hall, New Jersey 1995.

Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., New York 1989.

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Fermat's Last Theorem

Fermat's (pronounced Fer mah') Last Theorem is the name given to a puzzling problem that Pierre Fermat (1601-1665), a French amateur mathematician, created. Although he made contributions to several branches of math, his preferred area was in number theory - the relationships that exist among numbers.

It was while he was studying a copy of an ancient work on number puzzles that he started pondering about Pythagoras' (pronounced Pi tha' gor us) equation about right triangles. Pythagoras, a Greek mathematician who lived more than 2000 years before Fermat, proved that squaring the lengths of the two legs of a right triangle and adding them together would be equal to the square of the length of the longest side, usually stated as a2 + b2 = c2.

As Fermat meditated on this equation, he began speculating on a related one, a3 + b3 = c3. Through trial and error he soon saw that finding integers that made this equation true seemed to be impossible. After trying powers of 4 and 5 and being equally frustrated, he wrote in the wide margin of that ancient text that there are no numbers that will make the general equation

an + bn = cn, when 'n' is an integer greater than 2.

The genius then mischievously added, "I can prove this but don't have enough room to write it in the margin."

For 300 years mathematicians have tried to find the proof or explanation that Fermat didn't write, Fermat's Last Theorem. The American mathematician, Andrew Wiles, created a proof, though it took him eight years. The proof was published in Annals of Mathematics in 1995, 330 years after Fermat's death, but obviously it was not Fermat's proof, since Wiles used math not available in the 17th century. Will someone discover Fermat's proof? That question remains unanswered.

Contributed by Charlene Evans


Singh, Simon. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Doubleday, 1997.

Ball, W.W. Rouse. A Short Account of the History of Mathematics. 1908.

Web address for more information.

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What’s so magic about magic squares,

Originally, magic squares were of little mathematical significance when they were discovered around 2000 BC by Yu the Great, a Chinese emperor. During the reign of Yu the Great, he encountered two animals that were said to have magical powers, a tortoise and a dragon horse. On the back of the tortoise was a pattern or symbol, now known as the lo shu, which represented a square array of numbers. These numbers were first depicted by knots tied in strings, with white knots for odd numbers and black knots for even.

These mystical squares found their way to Japan, India, and the Middle East. There is evidence of magic squares being found in Arabia and even in Hebrew writings. So, why all the magic connected with these squares?

The ancient people thought the squares were magic, or at least possessed some sort of good luck, because the sums of the numbers in each row, in each column and in each major diagonal was the same. Silver plates used to be engraved with magic squares and prescribed as a charm against the plague and other serious illnesses. They were also connected to alchemy and astrology and were used by fortune tellers to decorate bowls and amulets.

During the thirteenth century, a mathematician name Yang Hui, actually began to study magic squares, referred to as vertical and horizontal diagrams. He extrapolated the ideas to construct a magic circle where the sum of the numbers on any diameter or along any of the concentric circles would be the same sum. Yang Hui also generalized some rules for constructing magic squares. In a 4X4 square there are 16 squares. Using the numbers 1 through 16, fill in the matrix starting with upper left corner and counting down. Number five will be next to number 1. If the numbers at the corners of the inner square and the outer square are transposed, or swapped, a magic square will result with all rows, columns and diagonals adding up to 34.

Albrecht Durer is credited with showing the first magic square in print. This magic square appeared in an engraving and was a 4X4 square with all rows, columns and diagonals summing to 34. In the bottom row the numbers appear in the following order: 4, 15, 14, 1. The year that the engraving was made was worked into these numbers. Can you guess the year of the engraving? The engraving’s date was 1514.

Durer's Magic Square

Magic Squares were studied in France and in Poland in the seventeenth century and methods of constructing magic squares were revealed. Three dimensional magic squares were also constructed by Adamas Kochansky in 1686.

During the nineteenth century, magic squares were taken seriously and were applied to problems dealing with probability and analysis. A type of magic square, developed by Euler, referred to as the Greco-Latin square, had use in designing experiments.

As recently as 1978, Allen Adler, a mathematician, has published papers about magic squares. His opinion is that most mathematicians view magic squares as a waste of valuable time because entertainment or recreation is the most that one could get from reaching a solution. However, he is convinced that the use of magic squares in the classroom could teach factorization into primes and association. Allen Adler has a strong message that reaches way beyond the topic of mathematics and the notion of magical experiences. On his web page that is dedicated to magic squares, there is an autobiography that reveals the course of study and exploration that has progressed. Of his experience he writes:

“I can trace the precise evolution of these ideas and results, but what is more important for students and others, . . . unlike the ancient methods and results for magic squares, which one finds repeated over and over again down the centuries, what one finds here is alive and growing. It also serves to show that anyone can follow the path to original discoveries by following his or her curiosity wherever it leads, and this is a lesson that needs very much to be communicated.”
Perhaps there certainly was magic in these squares for Mr. Adler. But you won’t find any amulet around his neck. Link to his web page to learn more:

Contributed by Cynthia Schmidt


  1. The National Council of Teachers of Mathematics. (1969).
  2. Historical Topics for the Mathematics Classroom. Washington, DC: Heck, W. and Fey, J.

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Moessner's Magic

Number sequences can be very interesting and thought provoking. Most number sequences involve building some kind of pattern and seeing what comes next. One well-known number sequence is Pascal's triangle. A number sequence not so well-known is Moessner's magic.

Let's consider the sequence of counting numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, . . . .

Now, cross out every second number, leaving

1, 3, 5, 7, 9, 11, 13, 15, . . . ;

form the cumulative totals of these numbers: 1, (1 + 3), (4 + 5), (9 + 7), (16 + 9), (25 + 11), (36 + 13), (49 + 15) . . . ;

and you get the sequence of consecutive squares:

1, 4, 9, 16, 25, 36, 49, 64, . . . !

This transformation of the sequence of counting numbers into another was first discovered and explored by Alfred Moessner in the early 1950s.

Now, suppose we start with the sequence of counting numbers just like before, but now we take out every third number and add what's left.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, . . .

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, . . .

1, (1 + 2), (3 + 4), (7 + 5), (12 + 7), (19 + 8), (27 + 10), (37 + 11), (48 + 13) + (61 + 14) ...

1, 3, 7, 12, 19, 27, 37, 48, 61, 75, . . .

Now, remove every second number in the new list and add the remaining numbers.

1, 7, 19, 37, 61, . . .

1, (1 + 7), (8 + 19), (27 + 37), (64 + 61), ...

1, 8, 27, 64, 125, . . .

Amazing! This is the sequence of cubes. Now, if you go through the same procedure again, this time crossing out every fourth number at the beginning, the result should not be surprising. You end up with the sequence of fourth powers: 1, 16, 81, 256, . . . . In general, taking out the nth number to begin with gives a resulting sequence of nth powers.

Try it with higher powers. It will work.

Contributed by Amy Troutman


  1. Conway, John H.and Richard K. Guy. The Book of Numbers. New York: Copernicus, 1995.
  2. URL 6/10/00 5:00pm

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