holds that . That is, the order of a unit in the ring of integers modulo divides and Minimality Suppose for all numbers coprime with . Then . Proof: If with , then for all numbers coprime with . It follows , since and the minimal positive such number. divides This follows from elementary group theory, because the exponent of any finite group must divide the order of the group. is the exponent of the multiplicative group of integers modulo while is the order of that group. In particular, the two must be equal in the cases where the multiplicative group is cyclic due to the existence of a primitive root, which is the case for odd prime powers. We can thus view Carmichael's theorem as a sharpening of Euler's theorem. Divisibility Proof. By definition, for any integer , we have that , and therefore . By the minimality property above, we have . Composition For all positive integers and it holds that . This is an immediate consequence of the recursive definition of the Carmichael function. Exponential cycle length If is the biggest exponent in the prime factorization of , then for all (including those not coprime to ) and all , In particular, for square-free (), for all we have Extension for powers of two For coprime to (powers of) 2 we have for some . Then, where we take advantage of the fact that is an integer. So, for , an integer: By induction, when , we have It provides that is at most . Average value For any : (called Erdős approximation in the following) with the constant and , the Euler–Mascheroni constant. The following table gives some overview over the first values of the function, for both, the exact average and its Erdős-approximation. Additionally given is some overview over the more easily accessible with . There, the table entry in row number 26 at column → 60.49 indicates that 60.49% (≈ ) of the integers have meaning that the majority of the values is exponential in the length of the input , namely {| class="wikitable" style="text-align:right" |- style="vertical-align:top" ! || || sum || average || Erdős average || Erdős /exact average || average || % > || % > |- |5||31||270||8.709677||68.643||7.8813||0.678244||41.94 ||35.48 |- |6||63||964||15.301587||61.414||4.0136||0.699891||38.10 ||30.16 |- |7||127||3574||28.141732||86.605||3.0774||0.717291||38.58 ||27.56 |- |8||255||12994||50.956863||138.190||2.7119||0.730331||38.82 ||23.53 |- |9||511||48032||93.996086||233.149||2.4804||0.740498||40.90 ||25.05 |- |10||1023||178816||174.795699||406.145||2.3235||0.748482||41.45 ||26.98 |- |11||2047||662952||323.865169||722.526||2.2309||0.754886||42.84 ||27.70 |- |12||4095||2490948||608.290110||1304.810||2.1450||0.761027||43.74 ||28.11 |- |13||8191||9382764||1145.496765||2383.263||2.0806||0.766571||44.33