Aviva Trivia

Prime numbers are integers that have no divisors except themselves and one.  The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, … and there are infinitely many. Below are ten examples of prime numbers that have unusual properties that you will not learn about in school. Tell your friends or your next Saturday-night date about them and they will surely be impressed with your arcane mathematical knowledge.

379009. When you type this prime into a calculator and turn it upside down it spells ˜Google,’ the name of the popular web search engine.

613. This is a very unusual prime and one of my favorites. You can get three different classes of numbers with this prime by simply rearranging its digits. If you move the first digit to the end you get 136, which is a triangular number (they have the formula  n * (n + 1)/2 and (16 * 17)/2 = 136), and if you move the first digit of 136 to the end you get 361, which is a square: 19^2=361.

433.  This prime is close to the title of a composition by avant-garde composer John Cage. His piece 4’33” (referred to as “four thirty-three”) entails playing nothing at all for four minutes and thirty-three seconds.

81457.  Leetspeak is a kind of coded language used among a segment of the Internet population. If we use a portion of the Leet cipher: 8 = b, 1 = l, 4 = a, 5 = s, 7 = t, the prime 81457 spells the word “blast” in Leetspeak. 

77345993. Another calculator prime. When turned upside down it looks as if the word EGGSHELL is spelled.

6089. This is the beginning of a curious sequence of six primes whose digits contain only circles (0, 6, 8, 9). They have the form 609 * 10^n- 1. They can be arranged nicely in a stack like this:

6089
60899
608999
6089999
60899999
608999999

6823. According to the Jewish Encyclopedia, the Tetragrammaton (YHWH) is one of the names of the God of Israel and it occurs 6823 times in the Old Testament.

15251. This isn’t a prime but it has some curious properties related to them. It’s the least palindromic number (it reads the same way forward and backward) such that the sum of primes from its smallest to largest prime factor is also a prime, and the sum of composites from its smallest to largest prime factor is a prime. I.e., 15251 = 101 * 151, and  101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 = 1367, a prime; and the sum of all composites between 101 and 151:  102 + 104 + 105 + 106 + … (there are a lot of them) … + 147 + 148 + 150 = 5059, a prime.

(6 * 10^(6254+10) + 6881691889) * 10^(6254+1) + 9. This is the expression for a probable prime that has 12,520 digits.  It is called a ˜probable prime’ because although it has passed Fermat tests (I won’t explain those here) no one is sure how to certify the number prime because it doesn’t have an easily provable form. It’s also a ˜strobogrammatic’ prime because it will look the same when rotated 180 degrees: The digits 0, 1, and 8 look the same when turned upside down and the digits ˜6′ and ˜9′ are considered vertical reflections of each other.

(4 * 10^(5819+13) + 3141592653589) * 10^(5819+1) + 3. This is another probable prime and it has 11,653 digits. Notice that the first 13 digits of Pi = 3.141592653589793 … occur in its center. Strange.

And there you have ten unusual and wonderful prime numbers. Memorize their properties and tell your family and friends about them!

Two famous unsolved problems in number theory are 1) Goldbach’s Conjecture, which claims that every even integer > 2 is expressible as the sum of two primes; and 2) de Polignac’s conjecture, which claims that every even integer can be expressed as the difference of two consecutive primes in infinitely many ways. As these famous problems suggest, it’s interesting to observe the various ways in which numbers can be “decomposed” into prime numbers, as well as into other classes of numbers.

The purpose of this article is to investigate a somewhat unusual decomposition problem. Positive integers will be tested to see if they can be expressed as the sum of a Fibonacci number and a prime number, respectively. We will refer to these numbers as Fibonacci Prime Decompositions (FPDs).

For example, 122 is a FPD because it can be expressed thus:

122 = 13 + 109 = 21 + 101 = 55 + 67.

Note that the first number in each of the three summations is a Fibonacci number, while the second is a prime (13 is both). (A Fibonacci number is defined as: Start with 0 and 1, then get the next terms by adding the previous two Fibonacci numbers. The sequence starts: 0, 1, 1, 2, 3, 5, 8, 13, 21, …  A prime is a number whose only divisors are itself and 1.)

Let W(n) denote the number of ways an integer n can be expressed as the sum of a Fibonacci number and a prime. Our first sequence is of W(n) for n from 1 to 25.

Sequence 1:
     n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, …
W(n): 0, 1, 2, 2, 2, 2, 2, 2, 1,  3,   2,   2,   3,   2,   2,   2,  1,   2,   3,   2, …

More zeroes eventually occur in Sequence 1, which means that there are numbers not expressible as the sum of a Fibonacci number and a prime. The following sequence is of the n such that W(n)=0.
Sequence 2:
1, 35, 119, 125, 177, 208, 209, 221, 255, 287, 299, 329, 363, 416, 485, 515, 519, 535, 539, 551, 561, 567, 637, 697, 705, 718, 755, 768, 779, 784, 793, 815, 869, 875, 899, 925, 926, …

Conjecture:  There are infinitely many odd as well as even integers that cannot be expressed as the sum of a Fibonacci number and a prime.

A prime p will never occur in Sequence 2 since  p + 0  will always be at least one representation.
 
Unsolved question: Will a Fibonacci number > 1 ever be present in Sequence 2?

The following sequence is the least number k such that it has n unique representations as the sum of a Fibonacci number and a prime.

Sequence 3:

    n                 k
             1                 3
         2                 4
                      3                 8
    4                 24
    5                 74
    6                 444
    7                 1614
    8                 15684
For example, 15684 is the least FPD number having eight representations because:

    15684 = 1 + 15683
    15684 = 5 + 15679
    15684 = 13 + 15671
    15684 = 55 + 15629
    15684 = 233 + 15451
    15684 = 377 + 15307
    15684 = 1597 + 14087
    15684 = 4181 + 11503
Unsolved questions: What is the least FPD having 9 representations? Why do so many of the k values in Sequence 3 end in the digit 4?

John Forbes Nash Jr. worked with Goldbach decompositions (his short note on it can be found at his Princeton web site under the section “Goldbach Programs”) and he noticed that it’s sometimes possible to find even integers that can be expressed as a sum of two primes, such that the smaller prime cubed is greater than the larger prime. Taking inspiration from Nash’s observation, I examined FPDs such that W(n)=1, and noticed that some of the decompositions were such that the prime was less than the square root of the Fibonacci number. For example, 155 can only be expressed as 155 = 144 + 11; and notice that sqrt(144) = 12 > 11. A computer search of all n <= 105  revealed that integers with this property seemed to be rare. The following FPDs were found. Sequence 4: 146, 155, 629, 1599, 2615, 2631, 4183, 4186, 4192, 10963, 10969, 10977, 10999, 11017, 11019, 11025, 46375, 46379, 46387, 46391, 46397, 46409, 46421, 46429, 46435, 46469, 46565, 46579, 75028, 75036, 75288, ... Conjecture:  There are only finitely many FPDs with exactly one representation, such that in their decomposition the prime is less than the square root of the Fibonacci number. Another unusual yet interesting sequence is those FPDs with exactly one representation such that both numbers have the same number of decimal digits. Sequence 5: 2, 9, 65, 77, 93, 95, 123, 323, 335, 343, 377, 395, 415, 425, 437, 527, 545, 553, 583, 586, 670, 700, 715, 723, 726, 731, 749, 783, 801, 804, 833, 838, 849, 851, 901, 903, 905, 906, 923, 957, 959, 964, 965, 1003, 1078, 1081, 1113, 1115, ...  For example, in the decomposition of 1081 the prime and Fibonacci number both have three digits: 1081 = 144 + 937. Unsolved question: Is Sequence 5 infinite?

The decimal expansion of p begins 3.14159265358 … and the number is defined as the ratio of a circle’s circumference to its diameter. If you fast forward to about the 760th digit in the decimal expansion of p, you will see the string “…34 999999 83…”  That group of six repeating 9s in the center is known as the ‘Feynman Point.’

p is a real number that is irrational and transcendental, meaning it is not the ratio of two integers and it is not the solution of a polynomial with rational coefficients. Although it hasn’t  been proved, it’s generally believed that any finite set of digits will eventually occur in the decimal expansion of any irrational number due to their non-repeating behavior. For example, it’s thought that at some point in the decimal expansion of, say, v666 that the string ‘6660066600666’ will occur, and the string ‘12345432123454321’ will occur, and any other finite string you come up with. Thus, the appearance of six 9s in the decimal expansion of p isn’t that unusual. But the fact that it occurs after only 762 digits is what makes it a genuine curiosity. (For comparison, the earliest location of any four repeating digits is at position 1589, in which four 7s appear.)
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If you separate each of the digits of the number 8166 and form a sequence by summing them in a Fibonacci-like fashion thus: 8 + 1 + 6 + 6 = 21,  1 + 6 + 6 + 21 = 34, 6 + 6 + 21 + 34 = 67, …, after 14 terms you will see the reversal of 8166 appear. Here is the full sequence: 8, 1, 6, 6, 21, 34, 67, 128, 250, 479, 924, 1781, 3434, 6618.

I have dubbed numbers with this property “revrepfigits,” (reverse replicating Fibonacci-like digits) after “repfigit” numbers (defined below). A formal definition of revrepfigits is, Numbers n such that their reversal occurs in a sequence generated by starting with the n digits of a number and continuing the sequence with a number that is the sum of the previous n terms.

Revrepfigits are similar to repfigits, which were introduced by Mike Keith in 1987. Repfigits are numbers having the same property as revrepfigits except the original number appears in the generated sequence instead of its reversal. For example, 3684 is a repfigit since it occurs in the sequence 3, 6, 8, 4, 21, 39, 72, 136, 268, 515, 991, 1910, 3684. The word Fibonacci in the definition comes from the well-known sequence defined as Fn = Fn – 1  + Fn – 2, F0 = 0, F1 = 1, F2  = 1, (where each term is the sum of the previous two) which produces: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …
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A palindrome is a number that reads the same way forward and backward. Examples are 1234321, 484, and 6300036. A number’s divisors are all the numbers that divide evenly into it. The divisors of 10, for example, are 1, 2, 5, and 10 because 10/1 = 10, 10/2 = 5, 10/5 = 2, and 10/10 = 1. Let’s combine the concepts of palindromes and divisors: What would the sequence of numbers that possess an abundance of palindromic divisors look like?

1, 2, 4, 6, 12, 24, 66, 132, 264, 792, 1848, 2772, 5544, 13332, 14652, 24024, 26664, 72072, 79992, 186648, 205128, 264264, 559944, 792792, 1333332, 2666664, 7279272, 7999992, 13333320, 14666652, 26690664, 29333304, 80071992, 134666532

A formal definition of this sequence is: Numbers n such that the amount of palindromic divisors of n sets a new record. In Clifford A. Pickover’s book, A Passion For Mathematics, he dubbed these numbers “palindions” and wrote: “Palindions are natural numbers that have more palindromic divisors than any smaller number” (p. 107). For example, 2666664 is a palindion because it has exactly 50 palindromic divisors, and no smaller number has that many. (The sequence above is listed as A093036 in Neil Sloane’s Online Encyclopedia of Integer Sequences.)
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It’s an uncertain world. No one knows from this day to the next what will happen or what to expect. Life itself is a gamble, and we play the odds everyday, irregardless of who those odds favor. However, we still want to know if the odds are in favor or against a certain event, such as winning the lottery. In fact, it’s about three times more likely for you to be killed while driving to the store to buy the lotto ticket than it is to win the actual jackpot. You also have a higher chance of being struck and killed by lightning than you do of becoming multi-millionaire.

With yet another Final Destination movie out, the idea of death and how we may die is ingeniously implanted within our minds. We are left to ponder and question the odds of our very existence. We all know one thing for sure; the odds of dying are 100 percent; there is no way to cheat death, as the movie might lead one to believe. But what are the odds that death will occur at any given time? There are no absolute odds; the odds of dying from various causes are unique to each individual. We can easily determine the approximate chances based on the actual deaths occurring each year divided by the total population; however, these odds are greatly affected by the activities with which people take part in, the areas with which they live and drive, the type of work they do, and many other factors. The odds of being in the wrong place at the wrong time are about a million to one.
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Square numbers. 12  = 1, 22  = 4, 32  = 9, 42  = 16, 52  = 25, 62  = 36 … They are one of the most basic classes of numbers. Many people consider a geometric square to be a perfect shape with the most pleasing type of symmetry, which brings us to the reason numbers of the form n2 are called squares (sometimes ‘perfect squares’) – they have this simple yet elegant geometrical representation:

                                 * * * * * 
                      * * * *   * * * * *       
              * * *   * * * *   * * * * *
        * *   * * *   * * * *   * * * * *
    *   * *   * * *   * * * *   * * * * *   
    1    4      9        16         25      …
Square numbers have been studied for hundreds of years and many intriguing facts about them have been discovered. Below are a few properties I think are among the best.

1. A pattern often listed in number theory books is this one involving the sum of odd numbers to produce perfect squares:

                             1 =  1
                 1 + 3 =  4
             1 + 3 + 5 =  9
             1 + 3 + 5 + 7 = 16
             1 + 3 + 5 + 7 + 9 = 25
        1 + 3 + 5 + 7 + 9 + 11 = 36
                    …

It was first discovered by the Pythagoreans in ancient times and later appeared in Leonardo Fibonacci’s book Liber quadratorum (Book of Squares) in 1225. It is fairly easy to prove with induction that the pattern will continue to hold, but I will omit the proof here for space (and technical) reasons.

2. A mathematician known only as Dr. Hutton published the first table of squares up to 25,400 in the year 1781. After learning of the table, mathematicians and amateurs gleaned many facts from it, such as the simple observation that squares always end in the digits 0, 1, 4, 5, 6, or 9.

3. If you square nine ones after they have been concatenated together, you will get this nice palindrome (which is a number reading the same way forward and backward):

1111111112 = 12345678987654321

4. A number that undulates is one that has the digital pattern abababab… where its digits rise and fall in equal steps. For example, 282828 and 919191 are undulating numbers.

The square 2642 = 69696 is the largest known undulating square. None larger has ever been found, even though squares with millions of digits have been tested by computer. However, if one considers numerical bases other than base-10, undulating squares can be found. One example is 2922 = 85264, which becomes 41414 when written in base-12.

5. A famous unsolved problem in number theory is whether or not there are any more solutions to x! + 1 = y2 other than x = 4, 5, and 7 (where x! is the factorial of x). For example,

4! + 1 = 25 = 52
5! + 1 = 121 = 112
7! + 1 = 5041 = 712

This is known as Brocard’s Problem and it has been checked via computer up to x = 109, with no other solutions found.

6. Curious mathematicians, programmers, and amateurs have also wondered if squares can be composed only of certain digits. Here is one with only the digits 1, 4, and 9:

6480702115891070212 = 419994999149149944149149944191494441

7. Joseph Louis Lagrange proved in 1770 that every positive integer can be expressed as the sum of not more than 4 positive squares. For example, the prime 31 can be written as

31 = 52  + 22 + 12 + 12
 
8. Here are the two largest and smallest perfect squares that use all the digits from 0 to 9:

990662 = 9814072356
320432 = 1026753849

9. Joseph Madachy discovered these numbers that are equal to their halves squared when they are split apart in this fashion:

1233 = 122 + 332
8833 = 882 + 332
5882353 = 5882 + 23532

10. Now we will end with a more advanced fact about squares which first appeared as a problem posed in the journal, American Mathematical Monthly, in the year 1957. The problem was titled, “Conjecture on Reversals” and involved squares along with reversing the digits of a number. To understand the problem, consider these products:

169 * 961 = 162409 = 4032
1089 * 9801 = 10673289 = 32672

Notice that we have multiplied 169 and 1089 by their reversals to get perfect squares. Also notice that both of the numbers we multiplied, along with their reversals, are squares themselves: 132 = 169, 312 = 961, 332 = 1089, and 992 = 9801.

The original conjecture published in the American Mathematical Monthly can be paraphrased as, ‘When a number and its reversal are not equal, only if both are perfect squares will their product also be a square.’

But running a computer search reveals this conjecture to be quite false. Here are just a few of many counterexamples found:

288, 528, 768, 825, 867, 882, 1584, 2178, 4851, 8712, 10989, 13104, 14544, 15984, 20808, 21978, 26208, 27648, 27848, 36828, 40131, 44541, 48139, 48951, 49686, 57399, 68694, 80262, 80802, 82863, 84672, 84872, 87912, 93184, 98901, 99375, 109989, …

For example, notice that 288 is not a perfect square: sqrt(288) = 16.97056…  nor is its reversal: sqrt(882) = 29.69848…, also notice that the number is not equal to its reversal, yet 288 * 882 = 254016 = 5042. 

So it seems that squares are not so easily tamed!

(The counterexamples above are listed as sequence A082994 in the Online Encyclopedia of Integer Sequences.)

And there you have ten enthralling facts about square numbers.