Fibonacci Prime Decompositions
by Jason Earls, author of Heartless Bastard In Ecstasy & Zombies of the Red Descent http://becomeguitaristfromhell.blogspot.com/ http://www.youtube.com/user/zevi35711
Two famous unsolved problems in number theory are 1) Goldbach’s Conjecture, which makes the claim that every even integer > 2 is expressible as the sum of two primes. 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 is interesting for mathematicians to categorize and calculate various ways in which numbers can be “decomposed” into prime numbers and other classes of integers. The purpose of this article is to investigate another (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; and 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 number (although 13 is actually both). Recall that a Fibonacci number is defined as: Starting with 0 and 1, produce the next term by adding the previous two Fibonacci numbers, and continue the process. Hence, the sequence starts: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...; and recall that a prime number is an integer 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 number, respectively. 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, ...
When the sequence is continued more zeroes eventually occur, which tells us 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 1: There are infinitely many odd as well as even integers that cannot be expressed as the sum of a Fibonacci number and a prime. Note that primes will never occur in sequence 2; and a (trivial) proof of this statement is: Since p + 0 will always make up at least one representation, W(p) >= 1. Unsolved question 1: Will a Fibonacci number > 1 ever be present in Sequence 2? Our next 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 1 2 3 4 5 6 7 8 k 3 4 8 24 74 444 1614 15684
For example, 15684 is the least FPD number having eight unique representations due to the following sums: 15684 15684 15684 15684 15684 15684 15684 = = = = = = = 1 + 15683 5 + 15679 13 + 15671 55 + 15629 233 + 15451 377 + 15307 1597 + 14087
15684 = 4181 + 11503 Unsolved questions 2 & 3: What is the least FPD having 9 representations? Why do so many of the k values in Sequence 3 end in the digit 4? The famous mathematician and Nobel Prize winner, John Forbes Nash Jr., worked with Goldbach decompositions (his short note on the problem can be found at his Princeton web site under the section “Goldbach Programs”) and Nash noticed that it is occasionally 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 <= 10 5 revealed that integers with this property were somewhat rare; and 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 2: 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 having exactly one representation such that both numbers have the same number of decimal digits, which gives us our next sequence. 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 4: Is Sequence 5 infinite? -end-
(Thanks for reading. If you now of any magazines that would like to publish this article, please contact the author: zevi_35711@yahoo.com. Also, you would be helping out the author greatly if you purchased one of his books from Amazon.com or another online book store of your choice. Thanks again.) http://becomeguitaristfromhell.blogspot.com/ http://zombiesofthereddescent.blogspot.com/ http://www.youtube.com/user/zevi35711 Bio: Jason Earls is the author of Cocoon of Terror (Afterbirth Books), Heartless Bastard In Ecstasy, How to Become a Guitar Player from Hell, Zombies of the Red Descent, If(Sid_Vicious == TRUE && Alan_Turing == TRUE) {ERROR_Cyberpunk(); }, Red Zen, and 0.136101521283655... all available at Amazon.com and other online book stores. His fiction and mathematical work have been published in Red Scream, Yankee Pot Roast, M-Brane SF, Scientia Magna, three of Clifford Pickover’s books, Mathworld.com, AlienSkin, Recreational and Educational Computing, Escaping Elsewhere, Neometropolis, Thirteen, Dogmatika, Prime Curios, the Online Encyclopedia of Integer Sequences, OG’s Speculative Fiction, Nocturnal Ooze, Bust Down the Door and Eat All the Chickens and other publications. He currently resides in Oklahoma with his wife, Christine.