arXiv:1103.3578v1 [math.NT] 18 Mar 2011 C sprime. is h ioac eune(e 9) rtesqec frepunit of sequence the sequen or certain in [9]), numbers (see Lehmer sequence known no Fibonacci is are number there the Lehmer that known No is base. it every in pseudoprimes hence, 1] 1] [14]. [13], [11], rmso hsfr.Ide,Hoe 8 hwdta o most for that showed [8] Hooley Indeed, that form. seems this it since of researchers primes of attention attracted They ry ec,if Hence, erty. any integer C hoe 1. Theorem o ninteger an For opst integer composite A oue0,Nme ,Pgs000–000 0002-9939(XX)0000-0 Pages S 0, Number 00, Volume SOCIETY MATHEMATICAL AMERICAN THE OF PROCEEDINGS lobe tde.Freape n[2 ti hw htfrmo for that shown is it [12] in example, For studied. been also ulnnmeswihaepedpie r eysac.A scarce. very are pseudoprimes are number which numbers Cullen o base a not φ i o unu n pseudo-prime any up turn not did ( n n m suethat Assume u euthr sta hr sn ulnnme ihteLehm the with number Cullen no is there that is here result Our A ULNNMESWT H EMRPROPERTY LEHMER THE WITH NUMBERS CULLEN ssur-re Write square-free. is scmoie o oeaottesting about more For composite. is g steElrfnto of function Euler the is ) ulnnumber Cullen ∈ oiebut posite the Abstract. m [2 sapstv integer positive a is , uhthat such n 00 se[].Frohrrslso emrnmes e [1] see numbers, Lehmer on results other For [4]). (see 1000] a hCle number Cullen th pedpie oecmue erhr pt eea millio several to up searchers computer Some -pseudoprime. JOS Let φ > a n ( φ MAR E ´ C ≥ ( C ee eso htteei opstv integer positive no is there that show we Here, C n n ) n 0 that 30, sanme fteform the of number a is ,a 1, a ) m | ethe be m | C C ´ AGA IA N LRA LUCA FLORIAN AND RIBAS GRAU IA 2. scle a called is n ≡ n pseudoprime − − ro fTerm1 Theorem of Proof a C 1. n 1. ,then 1, φ n (mod m m hCle ubr If number. Cullen th ( C = Introduction emrnmesaeCrihe numbers; Carmichael are numbers Lehmer . C n hc sabase a is which n n ) C 2 | = n emrnumber Lehmer m n C C a h rpryta ti com- is it that property the has 1 + 1 .PedpieCle ubr have numbers Cullen Pseudoprime ). Y oaybs.Tu,i ol emthat seem would it Thus, base. any to i n n =1 k obase to − sprime. is p ,btthat but 1, i C . n C o rmlt,se[]ad[6]. and [3] see primality, for n
c a XXAeia ahmtclSociety Mathematical American XXXX = a sacmoiiepositive compositive a is n φ suorm o any for pseudoprime if 2 ( C C n φ n n o some for 1 + ( ) m sntpie Then prime. not is | n ) ti adt find to hard is it C uhthat such | nbase in s n n m − h number the e,sc as such ces, Carmichael − st 1 although , then , ,where 1, n rprop- er , n C g [2], , ≥ n C for ns a 1. is n . 2 JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA
So, k n (pi − 1) | n2 . Yi=1 α n2 Write n = 2 n1, where n1 is odd. Then Cn = n12 + 1, where n2 := α + n. Let p be any prime factor of Cn. Since p − 1 | Cn − 1, it follows that np p = mp2 +1 for some odd divisor mp of n and some np with log n n ≤ n = n + α ≤ n + . p 2 log 2
Let us first show that in fact np ≤ n. Assume that np >n. Then, n (1) Cn = n2 +1= pλ, for some positive integer λ, where p ≥ 2n+1 +1. Observe that λ> 1 because Cn is not prime. Now C n2n + 1 λ = n ≤ < n. p 2n+1 + 1 Reducing equation (1) modulo 2n, we get that 2n | λ − 1, so 2n ≤ λ − 1
np Next we look at mp. If mp = 1, then p = 2 +1 is a Fermat prime. Hence, γp γp np = 2 for some nonnegative integer γ. Since 2 = np ≤ n, we get that γp < (log n)/(log 2). Hence, the prime p can take at most 1 + (log n)/(log 2) values. Next, observe that since
(2) mp | n,
pY|Cn it follows that the number of prime factors p of Cn such that mp > 1 is ≤ (log n)/(log 3). Hence, we arrived at the bound log n log n (3) k< 1+ + < 1 + 2.4 log n. log 2 log 3
We next bound np. Put N := ⌊ n/ log n⌋, and consider pairs (a, b) of integers in {0, 1,...,N}. There arep (N + 1)2 > n/ log n such pairs. For each 3/2 1/2 such pair, consider the expression L(a, b) := an + bnp ∈ [0, 2n /(log n) ]. Thus, there exist two pairs (a, b) 6= (a1, b1) such that 2n3/2/(log n)1/2 |(a−a )n+(b−b )n | = |L(a, b)−L(a , b )|≤ < 3(n log n)1/2. 1 1 p 1 1 n/ log n − 1
Put u := a − a1, v := b − b1. Then (u, v) 6= (0, 0) and 1/2 |un + vnp| < 3(n log n) . We may also assume that u and v are coprime, for if not, we replace the pair (u, v) by the pair (u1, v1), where d := gcd(u, v), u1 := u/d, v1 := v/d, 1/2 and the properties that max{|u1|, |v1|} ≤ (n/ log n) and |u1n + v1np| < CULLENNUMBERSWITH THE LEHMERPROPERTY 3
3(n log n)1/2 are still fulfilled. Finally, up to replacing the pair (u, v) by the pair (−u, −v), we may assume that u ≥ 0. n np Now consider the congruences n2 ≡ −1 (mod p) and mp2 ≡ −1 (mod p). Observe that 2, n, mp are all three coprime to p. Raise the first congruence to u and the second to v and multiply them to get
u v nu+npv u+v n mp2 ≡ (−1) (mod p). Hence, p divides the numerator of the rational number
u v nu+npv u+v (4) A := n mp2 − (−1) .
n2 Let us show that A 6= 0. Assume that A = 0. Recall that Cn = n12 + 1. Thus, expression (4) is
u v (n+α)u+npv u+v A = n1 mp2 − (−1) = 0. u v Then n1 mp = 1, (n + α)u + vnp = 0, and u + v is even. Since u ≥ 0, it follows that v ≤ 0. Put w := −v, so w ≥ 0. There exists a positive integer w u ρ which is odd such that n1 = ρ and mp = ρ . Since u and v are coprime and u + v is even, it follows that u and v are both odd. Hence, w is also odd. Also, since mp divides n1, it follows that u ≤ w. We now get α w (2 ρ + α)u − wnp = 0, so u w = α w . np 2 ρ + α α u α u The left–hand side is ≥ u/n = u/(2 ρ ), because np ≤ n = 2 ρ . Hence, we get that u u w u w w α u ≤ = α w leading to u ≤ w α ≤ w . 2 ρ np 2 ρ + α ρ ρ + (α/2 ) ρ For ρ ≥ 3, the function s 7→ s/ρs is decreasing for s ≥ 0, so the above inequality together with the fact that u ≤ w implies that u = w (so both are 1 because they are coprime), and that all the intermediary inequalities are also equalities. This means that u = w = 1, α = 0 and n = np, but all this is possible only when Cn = p, which is not allowed. If ρ = 1, we then get that n1 = 1, so every prime factor p of Cn is a Fermat prime. Hence, we get k γp γp n2 2 i P 2 i Cn = 2 +1= (2 +1) = 2 i∈I , Yi=1 I⊆{X1,...,k} and k ≥ 2, but this is impossible by the unicity of the binary expansion of Cn. Thus, it is not possible for the expression A shown at (4) to be zero. The size of the numerator of A is at most 1/2 1/2 1+|nu+npv| u |v| 1+3(n log n) 2(n/ log n) 2 n mp ≤ 2 n 1/2 1/2 1/2 < 21+3(n log n) +(2/ log 2)(n log n) < 26(n log n) . 4 JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA
In the above chain of inequalities, we used the fact that 3 + 2/ log 2 < 5.9, together with the fact that (n log n)1/2 > 10 for n ≥ 30. Thus, for n ≥ 30, we have that the inequality 1/2 (5) p< 26(n log n) holds for all prime factors p of Cn. Thus, we get the inequality k k n 6(n log n)1/2 6k(n log n)1/2 2
α β from [5]. Thus, n = 2 3 and so all prime factors p of Cn are of the form α1 β1 2 3 + 1 for some nonnegative integers α1 and β1. Now write k C − 1 1 (7) a = n = 1+ φ(C ) p − 1 n Yi=1 i for some integer a ≥ 2. Since 1 1+ < 1.46, 2α1 3β1 α1≥Y0, β1≥0 2α1 3β1 +1 prime we get that a < 2, which is a contradiction. This shows that in fact there are no numbers Cn with the claimed property. We end with some challenges for the reader.
Reearch problem. Prove that Cn is not a Camichael number for any n ≥ 1. If this is too hard, can one at least give a sharp upper bound on the count- ing function of the set C of positive integers n such that Cn is a Carmichael number? We recall that Heppner [7] proved that if x is large then the num- ber of positive integers n ≤ x such that Cn is prime is O(x/ log x), whereas in [12] it was shown that if a> 1 is a fixed integer then the number of positive integers n ≤ x such that Cn is base a-pseudoprime is O(x(log log x)/ log x). Clearly, imposing that Cn is Carmichael (which is a stronger condition) should lead to sharper upper bounds for the counting function of such in- dices n. Finally, here is a problem suggested to us by the referee. Theorem 1 shows that φ(Cn)/ gcd(Cn − 1, φ(Cn)) exceeds 1 for all n. Can one say something more about this ratio? For example, it is possible that a minor modification of the arguments in the paper would show that this function tends to infinity with n, but we have not worked out the details of such a deduction. It would be interesting to find a good (large) lower bound on this quantity which is valid for all n and which tends to infinity with n. How about for most n? What about lower and upper bounds on the average value of this function when n ranges in the interval [1,x] and x is a large real number? We leave these questions for further research. Acknowledgements. We thank the referee for a careful reading of the paper and for suggesting some of the questions mentioned at the end. F. L. was supported in part by grants PAPIIT 100508 and SEP-CONACyT 79685,
References [1] W. D. Banks, A. M. G¨uloglu and C. W. Nevans, ‘On the congruence n ≡ a (mod ϕ(n))’, INTEGERS 8 (2008), #A59. [2] W. D. Banks and F. Luca, ‘Composite integers n for which ϕ(n) | n − 1’, Acta Math. Sinica 23 (2007), 1915–1918. 6 JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA
[3] P. Berrizbeitia and J. G. Fernandes, ‘Observaciones sobre la primalidad de los n´umeros de Cullen’, Short communication in “Terceras Jornadas de Teor´ıa de N´umeros” (http://campus.usal.es/ tjtn2009/doc/abstracts.pdf). [4] J. Cilleruelo and F. Luca, ‘Repunit Lehmer numbers’, Proc. Edinburgh Math. Soc., to appear. [5] G. L. Cohen and P. Hagis, ‘On the number of prime factors of n if φ(n) | n − 1’, Nieuw Arch. Wisk. 28 (1980), 177–185. e 2 [6] J. M. Grau and A. M. Oller-Marc´en, ‘An O(log N) time primality test for gener- alized Cullen numbers’, Preprint 2010, to appear in Math. Comp. [7] F. Heppner, ‘Uber¨ Primzahlen der Form n2n + 1 bzw. p2p + 1’, Monatsh. Math. 85 (1978), 99–103. [8] C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press, Cambridge, 1976. [9] F. Luca, ‘Fibonacci numbers with the Lehmer property’, Bull. Pol. Acad. Sci. Math. 55 (2007), 7-15. [10] F. Luca, ‘On the greatest common divisor of two Cullen numbers’, Abh. Math. Sem. Univ. Hamburg 73 (2003), 253–270. [11] F. Luca and C. Pomerance, ‘On composite integers n for which φ(n) | n − 1”, Preprint, 2009. [12] F. Luca and I. E. Shparlinski, ‘Pseudoprime Cullen and Woodall numbers’, Colloq. Math. 107 (2007), 35–43. [13] C. Pomerance, ‘On the distribution of amicable numbers’, J. reine angew. Math. 293/294 (1977), 217–222. [14] C. Pomerance, ‘On composite n for which ϕ(n) | n − 1, II’, Pacific J. Math. 69 (1977), 177–186.
Departamento de Matematicas,,´ Universidad de Oviedo, Avda. Calvo Sotelo, s/n, 33007 Oviedo, Spain E-mail address: [email protected]
Instituto de Matematicas,,´ Universidad Nacional Autonoma de Mexico,,´ C.P. 58089, Morelia, Michoacan,´ Mexico´ E-mail address: [email protected]