Fibonacci Pseudoprimes and their Place in Primality Testing

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1 Fibonacci Pseudoprimes and their Place in Primality Testing Carly Allen December 2015 Abstract In this paper, we examine the basic building blocks of the Fibonacci Primality Theorem, as well as the theorem itself. Additionally, we see the power of the theorem s contrapostive statement as a primality test and other strong combinations of primality tests in mathematics. 1 Motivation Specifically, the motivation for this particular research stems from pure mathematical interest. What is the most efficient and accurate algorithm that detects primes? Does combining tests help? How can we quickly and accurately determine if an integer is prime? Additionally, this project, and primality testing in general, aids a branch of research called encryption. This branch of mathematics ensure internet security and is foundational for safe transactions on the internet. By ensuring a number is prime, you create stronger internet security. 2 Building Blocks To understand the purpose of primality testing, we must first understand basic elements of mathematics, the first being the definition of a prime. A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. Additionally, earlier factorization techniques must be considered. Trial division provides an unconditional, yet time-consuming algorithm. To trial divide n, up to n cases must be considered. This method, considered a factorization technique, requires more time and gives us more information than we are seeking. We aim to find primality tests that will simply call an integer prime (or probably prime) or composite. Next, we consider an early primality test that is derived from Fermat s Little Theorem. Fermat s Little Theorem, as a special case, says for any odd prime p, 2 p 1 1 (mod p). 1

2 After taking the converse of the theorem, we get the Fermat Base-2 Primality Test: For a given odd integer n > 1, if 2 n 1 1 (mod n), then n is composite. If the result is 1, call n probably prime. From here, we see our first example of a pseudoprime. Using the Base-2 Fermat test, every prime number will return probably prime. The test, though is not perfect and some composite numbers will additionally return probably prime. When this happens, such a number is considered a pseudoprime. As an example, take n = 341. Compute (mod 341). This number is equivelent to 1 (mod 341). By the Fermat Base-2 Primality test, this number would return probably prime, but we know 341 = 11 31, making 341 composite. Therefore, 341 is an example of a pseudoprime of the Fermat Base-2 Primality Test. Next we will define Fibonacci numbers. These are numbers in the Fibonacci sequence. The sequence F n of Fibonacci numbers is defined by the recurrence relation: F 0 = 0 and F 1 = 1 F n = F n 1 + F n 2 making the first few integers of the sequence as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... The final building block of Fibonacci Pseudoprimes that we will consider is the Legendre Symbol. The Legendre Symbol is defined as follows: Let p be an odd prime. For any integer m, the Legendre symbol ( p m ) is defined as follows: ( p ) m = +1 if m is quadratic residue modulo p. 1 if m is quadratic nonresidue modulo p. 0 if p divides m. This symbol simply detects primes. In the case where p = 5, the symbol will detect numbers which are 1 and 4 (mod 5) and assign them the number 1. We will see a direct use of this symbol in the Fibonacci Primality Theorem. 3 The Fibonacci Primality Theorem The Fibonnaci Primality Theorem stemmed from an observation of the Fibonnaci Sequence modulo prime numbers. Note the following examples: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, (mod 7): 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1,

3 (mod 11): 0, 1, 1, 2, 3, 5, 8, 2, 10, 1, 0, 1, 1, 2, 3, 5, 8... (mod 13): 0, 1, 1, 2, 3, 5, 8, 0, 8, 8, 3, 11, 1, 12, 0, 12, Observe that in the case of the sequence modulo 7, the 8th integer in the sequence is equivalent to 0 (mod 7). Similarly, it was noted that the 12th integer in the sequence modulo 13 is 0 (mod 13). There seems to be a pattern with the p+1 or p 1 term of the sequence modulo p. This discovery sparked the Fibonacci Primality Theorem. If n is a prime, then F n ( 5 0 (mod n), n) where ( ) 5 1 if n ±1 (mod 5). = 1 if n ±2 (mod 5). n 0 if n 0 (mod 5). Similar to the procedure with the Fermat Base-2 Test, we take the contrapositive of this theorem to create a primality test, which says that if F n ( 5 0 (mod n), n) then n is composite. If the result is 0, call n probably prime. When we take the integer n = 341, we will see that the Fibonacci Primality Test successfully returns composite, wheras the Fermat Base-2 Primality Test returned probably prime. This implies that 341 is pseudoprime in the context of the Fermat Base-2 Test, but is not an example of a pseudoprime for the Fibonacci Primality Test. We will see later the power of combining these tests. 4 Proof In essence, the proof is based on the idea that the Fibonacci sequence is an example of a Lucas Sequence. The Lucas Sequence may be understood as arithmetic in the ring R = Z[x]/ ( x 2 ax + b ). Two cases must then be considered. Case 1: x 2 ax + b is reducible modulo p. In this case, R = Z/(pZ) Z/(pZ), where R x = (p 1) 2. Case 2: ( x 2 ax + b is irreducible modulo p. In this case, R = F p 2, where R x = p 2 1 ) = (p 1) (p + 1). Put simply, these two cases take account for the ±1 cases of the Legendre Symbol. The cardinality of the group of units counts the number of noninvertible elements. In the first case, zero in the only element in Z/(pZ) that is not invertible. In the field with p 2 elements, zero is again the only non-invertible element. To prove concisely, we will look at the following proof. The Fibonacci Sequence satifies the recurrance u j = u j 1 + u j 2 with recurrance polynomial x 2 x 1. Consider more broadly the polynomial f(x) = x 2 ax + b where a,b Z, with =a 2 4b not a square. 3

4 Let U j = U j (a, b) = xj (a x) j x (a x) V j = V j (a, b) = x j (a x) j (mod f(x)) (mod f(x)) We take the remainder in Z[x] upon division by f(x). The sequences both satisfy the recurrence for the polynomial x 2 ax + b, namely, U j = au j 1 bu j 2, V j = av j 1 bv j 2 with initial values U 0 = 0, U 1 = 1, V 0 = 2, V 1 = a Since the sequence U j is constructed by reducing polynomials modulo x 2 ax + b, we are really just working with the polynomials in the ring R = Z n (x) / ( x 2 ax + b ) ( ). Suppose p is an odd prime with = 1. Then is not a square in Z p. Thus, R=Z p [x]/ ( x 2 ax + b ) is isomorphic to the finite field F p 2 with p 2 elements. The subfield Z p is recognized as those coset representatives i + jx with j = 0. In F p 2, the function σ that takes an element to its p-th power has the following properties σ(u + v) =σ(u) +σ(v) 2. σ(uv) =σ(u)σ(v) 3. σ (u) = u if and only if u is in the subfield Z p. We ve created the field F p 2 with p 2 elements and therefore provided roots for x 2 ax + b. The roots are x itself and a x, which are not in Z p. ( ) { x in the case = 1 : p a x (mod (f (x), p)) p (a x) p x (mod (f (x), p).) Using definitions, p x p+1 (a x) p+1 x (a x) (a x) x 0 (mod (f (x), p)). This implies. ( In the case where find that U p 1 0 p U p+1 0 (mod p) ) = 1, the ring is not a finite field. We will come to (mod p). This proves our theorem. 5 The Power of Combining Primality Tests Combining primality tests proves to be extremely powerful. Below you will find a table that compares the prevalence of pseudoprimes in several contexts. 4

5 In the second column, note the prevalence of Fibonacci Pseudoprimes. The third column shows the prevalence of Base-2 pseudoprimes and the final column gives the prevalence of their intersection. There is a clear advantage to combining the two tests, as their intersection is much smaller than the two sets individually. n fpsp s psp(2) s fpsp(2) s Dominic W. Klyve and Daniel Monfre Another example of a combination of primality tests is the BPSW Test. The BPSW (Baillie, Pomerance, Selfridge, Wagstaff) Test is a probabilistic primality testing algorithm that combines a base-2 Fermat test with a Lucas Probable Prime Test. No fpsp(2) s yet found are congruent to 2 or 3 modulo 5. Such a pseudoprime would be a counterexample to the BPSW and would be a BPSW pseudoprime, an example of which has yet to be found. Current fpsp s are ruled out by the BPSW Test. The BPSW test finds the first D in the sequence 5, -7, 9, -11, 13, -15,... for which the Jacobi symbol ( ) D n is 1. If it is a fibonacci pseudoprime, the test scans another integer in this sequence. No composite up to 2 64 passes the BPSW Test. This means that the test is deterministic up to The power of the BPSW test lies in the fact Fermat pseudoprimes and Lucas pseudoprimes share no known numbers. 6 References Baillie PSW primality test. (2015, November 28). In Wikipedia, The Free Encyclopedia. 5

6 Crandall, R., Pomerance, C. (2010). Prime numbers: A computational perspective (2nd ed.). New York City, NY: Springer. Goldmakher, L. (n.d.). Legendre, Jacobi and Kornecker symbols [White paper]. Retrieved December 12, 2015, from Williams College website: Klyve, D. W., Monfre, D.. Looking for fibonacci base-2 pseudoprimes. 6