Prime Recreations: An Olio of Curios about Prime Numbers

By Bruce Pyne

Bruce has created a work totally unique among books of this type. He chose to number his chapters with prime numbers and cover material not seen anywhere else. Particularly interesting are his chapters on digit patterns and primes forming triangles. Nearly half of the length of the book is devoted to eleven (a prime number) chapters of lists of primes. Here, the reader will find a delightful and motley mélange of unlikely subjects, many with clever titles, and all alphabetically arranged. Included are primes in sports, stock market primes, prime temperatures, and prime Zip codes. The final section of the book, which the author designates as Part C, contains a complete cross-reference of primes found in the book, additional tables, a glossary, a fairly extensive bibliography, and a multisection index. Any lover of primes and lists should add this book to their personal library.

• Language: English
• Publisher: Page Publishing, Inc.
• Release date: Jan 4, 2019
• ISBN: 9781642982916

Book preview

Chapter 2

Introduction

Prime numbers are those numbers which are divisible only by themselves and one. The first five are 2, 3, 5, 7, and 11. One simple way to describe what prime numbers are is the following: given fifteen pennies and asked to make a rectangle out of them, you would have little problem in creating an array three pennies high and five pennies long. However, if after that you are given two more, making seventeen, you would discover that you would be unable to form an array with no empty spots. This, in a nutshell, is the physical, tangible essence of primeness, what it means to be prime. Due largely to their randomness and unpredictability among the integers, these numbers have fascinated humans since the time of the ancient Greeks and have captivated this author for forty-five years.

For number enthusiasts everywhere, don’t ever get the idea that everything there is to discover about primes has already been discovered. There are plenty of things just waiting for someone to hit upon. And if it turns out that that something has already been discovered, you can invoke the saying Great minds think alike. Everyone should experience the thrill of discovering. The topics discussed in this book are fertile ground for further research. They may even suggest other areas not covered. Read, enjoy, ponder, and explore. Remember, it’s not work if you’re having fun. Make it a labor of love.

Generalizations about Primes

A handful of generalizations about prime numbers can be made.

All primes, except 2, are odd numbers.

All primes of two or more digits end in a 1, 3, 7, or 9. Any number ending in a 5 is a multiple of 5.

All primes, except 2, differ by 1 from a multiple of 4. Thus they take the form of either 4n – 1 or 4n + 1.

All primes of the form 4n + 1 may be expressed as the sum of 2 squares.

All primes, except 2 and 3, differ by 1 from a multiple of 6. Often, multiples of 6 are sandwiched between what are known as twin primes, those differing by 2. Note that (5, 7), (11, 13), and (17, 19) are pairs of twin primes.

The number of primes is infinite. There is no largest prime.

There is no formula that generates all primes.

With few special exceptions, no formula generates only primes.

Every prime number can be the short leg of a right triangle.

Every even number is expressible as the difference between two primes.

The Fundamental Theorem of Arithmetic

Prime numbers may be thought of as being the building blocks of the integers. The fundamental theorem of arithmetic states that, except for the number 1, all positive integers are either prime, a power of a prime, or the product of primes. Numbers, other than 1, which are not prime are said to be composite. Semiprimes are those numbers which are either the square of a prime or the product of two distinct primes. The list of semiprimes starts off with 4, 6, 9, 10, 14, and 15. Note that 14 and 15 are the first pair of semiprimes that are consecutive integers. The next pair is 21 and 22. The first triple of semiprimes is 33 = 3 × 11, 34 = 2 × 17, and 35 = 5 × 7. Here are more such triples:

It is not possible to have four semiprimes in a row because one of the four would have one or more of the primes raised to a power (most often the 2).

Conjectures about Primes

Countless conjectures about primes have been made throughout history. Many of them have turned out to be false. Perhaps foremost among these is the one made by the seventeenth-century French mathematician Pierre de Fermat. He suggested that all numbers of the form 2 (raised to the power of 2n) + 1 are prime, but in 1732, Euler discovered that when n = 5 (making 2n = 32 and the whole expression equal to 2³² + 1), the result is composite and divisible by 641.

One well-known conjecture that has never been rigorously proven but for which no counterexample has been found is Goldbach’s conjecture. It states that every even number, except 2, is the sum of two primes. In fact, many are the sum of two primes in several ways. For instance, 30 = 7 + 23 = 11 + 19 = 13 + 17. The first number that equals the sum of two primes in two ways is 10. Every number higher than 12 is the sum of two primes in at least two ways. The first number expressible in three different ways is 22, in four different ways is 34, in five ways is 48, in six ways is 60, in seven ways is 78, in eight ways is 84, in nine ways is 90, and in ten ways is 114. Between 100 and 150, only 122 and 128 are expressible in fewer than five ways.

It is believed that the number of pairs of twin primes is infinite, but no one has been able to conclusively prove this yet.

Every even integer is the difference between two consecutive primes in an infinite number of ways. For example, 6 = 29 – 23 = 37 – 31 = 53 – 47 = 59 – 53, etc.

This author asserts that every number greater than 3 is the midpoint between two primes. Observe that 4 is the midpoint between 3 and 5, 5 is the midpoint between 3 and 7, 8 is the midpoint between 5 and 11, and so on. (One of the appendices lists all numbers from 4 to 120 as the midpoint between two primes.) The author also suspects that all gaps between consecutive primes for any prime p is less than the square root of p plus the natural logarithm of p. The author conjectures that every prime is a factor of some Fibonacci number.

Fibonacci numbers are a sequence of numbers starting with 1 and 1. Every term after the second is the sum of the two previous terms, so the third term is 1 + 1 = 2, the fourth term is 1 + 2 = 3, the fifth term is 2 + 3 = 5, and so on. The sequence continues 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, and 987. Several terms are prime, including 2, 3, 5, 13, 89, and 233. Every third term is a multiple of 2 and, thus, even. Every fourth term is a multiple of 3, every fifth term is a multiple of 5, and every eighth term is a multiple of 7. As we continue to extend the sequence, we would discover terms that are multiples of any prime we can state no matter how big.

Generating Primes

While there is no formula that generates all primes, there is a very simple method for producing them. It was discovered by the ancient Greek mathematician Eratosthenes, and he referred to it as a sieve. What you do is write out all numbers up to a certain point. Then cross out all multiples of 2, leaving only the odd numbers. Then cross out all multiples of 3 that are not already crossed out. Repeat the process with multiples of 5 and 7 and as high as you wish to go. You only need to go as high as the square root of the largest number you originally wrote down. All the remaining numbers will be prime.

There is another process of generating primes discovered by an Indian student in 1934, which amazingly works. This method involves first creating a grid or table. In the first row, the numbers 4, 7, 10, 13, 16, etc. are placed. These same numbers form the leftmost column. The second row, beginning with 7, continues with 12, 17, 22, 27, 32, 37, and so forth. This second row becomes the second column. The third row, starting with 10 and 17, continues by adding 7 to each succeeding number. Each nth row is also the nth column. The grid looks like this:

The grid continues indefinitely to the right and lower. Note that the difference between successive terms in the top row is 3, for the second row is 5, for the third row is 7, and, in general, for the nth row is 2n + 1. For every number n that is not in the table, 2n + 1 is a prime number. The only prime it misses is 2. To illustrate, 1 is not in the table and 2 × 1 + 1 = 3, which is prime. The next number not in the table is 2 and 2 × 2 + 1 = 5, which is prime. Next is 3 and 3 × 2 + 1 = 7, which is prime. Try it yourself with 5, 6, 8, 9, 11, and 14. Do you get six more primes?

Conversely, doubling any number in the grid and adding one will always result in a composite number.

An Unexpected Connection

All primes of the form 4n + 1 can be expressed as the sum of two squares. The first one is 5 = 4 × 1 + 1 = 2² + 1². The next one is 13 = 4 × 3 + 1 = 2² + 3². After that, there’s 17, 29, 37, and 41. All these primes are the lengths of the hypotenuses in primitive Pythagorean triangles. The word primitive simply means that the sides of the triangle are not multiples of a similar smaller triangle. Thus a 3-4-5 triangle is primitive, but a 6-8-10 triangle is not. In addition, each of these primes are the midpoint between two odd squares. The following table should help to clarify this.

Columns A and B are the legs of Pythagorean triangles. Columns C and E are hypotenuses, all primes and all of the form 4n + 1. Columns D and F are odd squares. Column B is the difference between Columns E and D as well as between F and E. Column F is the sum of Columns B and C. Note that 5 is midway between 1 and 9, 13 is midway between 1 and 25, 17 is midway between 9 and 25, and so forth. The author does not remember ever seeing this described in any other book.

Chapter 3

Primes as Sums, Differences, and Midpoints

Primes of the Form 4n + 1 as the Sum of Two Squares

The following is a list of all prime numbers less than one thousand that are one more than a multiple of four:

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 181, 193, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 401, 409, 421, 433, 449, 457, 509, 521, 541, 569, 577, 593, 601, 613, 617, 641, 653, 661, 673, 677, 701, 709, 733, 757, 761, 769, 797, 809, 821, 829, 853, 857, 881, 929, 937, 941, 953, 977. Primes in bold are pairs of cousin primes (those that differ by 4).

In the table below, note that even numbers form the top row and odd numbers form the leftmost column. Every number in the table is a prime number that is the sum of the squares of an even number and an odd number. All such primes less than a thousand (those in the list above) are included. In order for this sum to be a prime, the odd and even numbers must not have any factors in common. The actual technical term is that the odd and even numbers are relatively prime and that their LCD, or least common divisor, is one. The numbers in the bottom row are simply the number of primes in each column.

Primes as the Sum of Two Consecutive Squares

In the list below, observe that, generally, the units’ digits of the primes alternate 1 and 3. Sometimes, the same square gets used twice as is the case with the squares of 5, 30, and 35.

Note that all the primes ending in the digits 13 are the sum of squares ending in 2 and 3 or 7 and 8. However, 1513, which equals 27² + 28², is composite (17 × 89). A few squares, such as those of 11, 16, 21, 24, 27, and 28, among others, don’t get used. Listed below are all primes of this form below 10,000.

5 = 1² + 2²

13 = 2² + 3²

41 = 4² + 5²

61 = 5² + 6²

113 = 7² + 8²

181 = 9² + 10²

313 = 12² + 13²

421 = 14² + 15²

613 = 17² + 18²

761 = 19² + 20²

1013 = 22² + 23²

1301 = 25² + 26²

1741 = 29² + 30²

1861 = 30² + 31²

2113 = 32² + 33²

2381 = 34² + 35²

2521 = 35² + 36²

3121 = 39² + 40²

3613 = 42² + 43²

4513 = 47² + 48²

5101 = 50² + 51²

7321 = 60² + 61²

Primes as the Midpoint Between Two Consecutive Odd Squares

Note that here all primes (in bold) except 5 end in either a 1 or a 7 and all are one more than an even square. Note that the difference between the midpoint prime and either square is the sum of the two square roots. To illustrate, 5 – 1 = 1 + 3 = 4; 17 – 9 = 3 + 5 = 8; 37 – 25 = 5 + 7 = 12; and so on.

1² = 1 – 5 – 9 = 3²

3² = 9 – 17 – 25 = 5²

5² = 25 – 37 – 49 = 7²

9² = 81 – 101 – 121 = 11²

13² = 169 – 197 – 225 = 15²

15² = 225 – 257 – 289 = 17²

19² = 361 – 401 – 441 = 21²

23² = 529 – 577 – 625 = 25²

25² = 625 – 677 – 729 = 27²

35² = 1225 – 1297 – 1369 = 37²

39² = 1521 – 1601 – 1681 = 41²

53² = 2809 – 2917 – 3025 = 55²

55² = 3025 – 3137 – 3249 = 57²

65² = 4225 – 4357 – 4489 = 67²

73² = 5329 – 5477 – 5625 = 75²

83² = 6889 – 7057 – 7225 = 85²

89² = 7921 – 8101 – 8281 = 91²

93² = 8649 – 8837 – 9025 = 95²

Consecutive Primes That Sum to a Square

These are a fairly rare breed, so there’s far fewer of them than in the type listed in the previous section. Perhaps the reader can find other examples of this type.

17 + 19 = 36 = 6²

47 + 53 = 100 = 10²

71 + 73 = 144 = 12²

283 + 293 = 576 = 24²

13 + 17 + 19 = 49 = 7²

37 + 41 + 43 = 121 = 11²

Primes as the Sum of Three Squares

Every integer may be expressed as the sum of at most four squares. Primes as the sum of three squares can take several forms that are not necessarily mutually exclusive.

A. As the sum of the squares of consecutive integers:

29 = 2² + 3² + 4²

149 = 6² + 7² + 8²

509 = 12² + 13² + 14²

677 = 14² + 15² + 16²

1877 = 24² + 25² + 26²

3677 = 34² + 35² + 36²

8429 = 52² + 53² + 54²

9749 = 56² + 57² + 58²

B. As the sum of consecutive odd squares:

83 = 3² + 5² + 7²

251 = 7² + 9² + 11²

683 = 13² + 15² + 17²

1091 = 17² + 19² + 21²

2531 = 27² + 29² + 31²

5051 = 39² + 41² + 43²

7211 = 47² + 49² + 51²

C. As the sum of the squares of three primes (not necessarily unique):

17 = 2² + 2² + 3²

59 = 3² + 5² + 5²

107 = 3² + 7² + 7²

179 = 3² + 7² + 11²

227 = 3² + 7² + 13²

347 = 3² + 7² + 17²

419 = 3² + 7² + 19² = 3² + 11² + 17²

467 = 3² + 13² + 17²

491 = 3² + 11² + 19²

587 = 3² + 7² + 23²

659 = 3² + 11² + 23² = 3² + 17² + 19²

827 = 3² + 17² + 23²

D. As the sum of the squares of three consecutive odd numbers ending in the same digit:

1523 = 11² + 21² + 31²

1787 = 13² + 23² + 33²

3083 = 21² + 31² + 41²

3467 = 23² + 33² + 43²

8627 = 43² + 53² + 63²

E. No particular pattern:

17 = 2² + 2² + 3²

41 = 1² + 2² + 6² = 3² + 4² + 4²

53 = 1² + 4² + 6²

59 = 1² + 3² + 7²

61 = 3² + 4² + 6²

Primes as the Sum of Three Consecutive Increasing Powers

Recall that any number to the zero power is 1 and any number to the first power (i.e., 1) is itself. These are yet another method of representing primes as a sum of three numbers.

7 = 1 + 2 + 2²

13 = 1 + 3 + 3²

31 = 1 + 5 + 5²

43 = 1 + 6 + 6²

73 = 1 + 8 + 8²

157 = 1 + 12 + 12²

241 = 1 + 15 + 15²

307 = 1 + 17 + 17²

421 = 1 + 20 + 20²

463 = 1 + 21 + 21²

601 = 1 + 24 + 24²

757 = 1 + 27 + 27²

1123 = 1 + 33 + 33²

1483 = 1 + 38 + 38²

1723 = 1 + 41 + 41²

Primes as the Sum of Two Powers

This represents more largely unexplored territory.

17 = 2³ + 3²

31 = 2² + 3³

41 = 2⁴ + 5²

43 = 2⁴ + 3³

59 = 2⁵ + 3³

73 = 2⁶ + 3²

89 = 2³ + 3⁴ = 2⁶ + 5²

97 = 2⁴ + 3⁴

113 = 2⁵ + 3⁴ = 2⁶ + 7²

137 = 2⁷ + 3² = 2⁴ + 11²

157 = 2⁵ + 5³

233 = 2⁶ + 13²

251 = 2³ + 3⁵

281 = 2⁸ + 5²

283 = 2⁸ + 3³

307 = 2⁶ + 3⁵

337 = 2⁸ + 3⁴ = 3⁴ + 4⁴

347 = 2² + 7³

353 = 2⁶ + 17²

359 = 2⁴ + 7³

521 = 2⁹ + 3²

593 = 2⁹ + 3⁴

599 = 2⁸ + 7³

617 = 2⁸ + 19²

641 = 2⁴ + 5⁴

733 = 2² + 3⁶

761 = 2⁵ + 3⁶

857 = 2⁷ + 3⁶

881 = 4⁴ + 5⁴

1033 = 2¹⁰ + 3²

1051 = 2¹⁰ + 3³

1367 = 2¹⁰ + 7³

1459 = 2⁷ + 11³

1753 = 2¹⁰ + 3⁶

Primes as the Sum of Three Cubes

Numerous primes can be represented as the sum of three cubes, occasionally in more than one way.

3 = 1³ + 1³ + 1³

17 = 1³ + 2³ + 2³

29 = 1³ + 1³ + 3³

43 = 2³ + 2³ + 3³

73 = 1³ + 2³ + 4³

127 = 1³ + 1³ + 5³

179 = 3³ + 3³ + 5³

251 = 1³ + 5³ + 5³ = 2³ + 3³ + 6³

277 = 3³ + 5³ + 5³

281 = 1³ + 4³ + 6³

307 = 3³ + 4³ + 6³

349 = 2³ + 5³ + 6³

359 = 2³ + 2³ + 7³

397 = 3³ + 3³ + 7³

433 = 1³ + 6³ + 6³

521 = 1³ + 2³ + 8³

547 = 2³ + 3³ + 8³

557 = 5³ + 6³ + 6³

577 = 1³ + 4³ + 8³

593 = 5³ + 5³ + 7³

701 = 4³ + 5³ + 8³

853 = 5³ + 6³ + 8³

857 = 4³ + 4³ + 9³

863 = 2³ + 7³ + 8³

953 = 2³ + 6³ + 9³

1009 = 4³ + 6³ + 9³ = 1³ + 2³ + 10³

Primes as the Difference of Two Powers

For this section, a table seemed like the easiest way to portray these representations. A perhaps surprising number of primes can be formed in this manner. The author was not able to find appropriate powers for certain primes such as 43, 59, 103, 163, and 167, among others, which is why there are no entries for these particular primes. A computer search might assist in filling these in. All the powers in this table were found using nothing more than a handheld Texas Instruments TI-30X calculator.

Column B = Column D and Column C = Column E

Column A is the prime

A = B – C = D – E

To clarify, 3 = 128 – 125 = 2⁷ – 5³