Misplaced Pages

Zsigmondy's theorem

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
On prime divisors of differences two nth powers

In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 {\displaystyle a>b>0} are coprime integers, then for any integer n 1 {\displaystyle n\geq 1} , there is a prime number p (called a primitive prime divisor) that divides a n b n {\displaystyle a^{n}-b^{n}} and does not divide a k b k {\displaystyle a^{k}-b^{k}} for any positive integer k < n {\displaystyle k<n} , with the following exceptions:

  • n = 1 {\displaystyle n=1} , a b = 1 {\displaystyle a-b=1} ; then a n b n = 1 {\displaystyle a^{n}-b^{n}=1} which has no prime divisors
  • n = 2 {\displaystyle n=2} , a + b {\displaystyle a+b} a power of two; then any odd prime factors of a 2 b 2 = ( a + b ) ( a 1 b 1 ) {\displaystyle a^{2}-b^{2}=(a+b)(a^{1}-b^{1})} must be contained in a 1 b 1 {\displaystyle a^{1}-b^{1}} , which is also even
  • n = 6 {\displaystyle n=6} , a = 2 {\displaystyle a=2} , b = 1 {\displaystyle b=1} ; then a 6 b 6 = 63 = 3 2 × 7 = ( a 2 b 2 ) 2 ( a 3 b 3 ) {\displaystyle a^{6}-b^{6}=63=3^{2}\times 7=(a^{2}-b^{2})^{2}(a^{3}-b^{3})}

This generalizes Bang's theorem, which states that if n > 1 {\displaystyle n>1} and n {\displaystyle n} is not equal to 6, then 2 n 1 {\displaystyle 2^{n}-1} has a prime divisor not dividing any 2 k 1 {\displaystyle 2^{k}-1} with k < n {\displaystyle k<n} .

Similarly, a n + b n {\displaystyle a^{n}+b^{n}} has at least one primitive prime divisor with the exception 2 3 + 1 3 = 9 {\displaystyle 2^{3}+1^{3}=9} .

Zsigmondy's theorem is often useful, especially in group theory, where it is used to prove that various groups have distinct orders except when they are known to be the same.

History

The theorem was discovered by Zsigmondy working in Vienna from 1894 until 1925.

Generalizations

Let ( a n ) n 1 {\displaystyle (a_{n})_{n\geq 1}} be a sequence of nonzero integers. The Zsigmondy set associated to the sequence is the set

Z ( a n ) = { n 1 : a n  has no primitive prime divisors } . {\displaystyle {\mathcal {Z}}(a_{n})=\{n\geq 1:a_{n}{\text{ has no primitive prime divisors}}\}.}

i.e., the set of indices n {\displaystyle n} such that every prime dividing a n {\displaystyle a_{n}} also divides some a m {\displaystyle a_{m}} for some m < n {\displaystyle m<n} . Thus Zsigmondy's theorem implies that Z ( a n b n ) { 1 , 2 , 6 } {\displaystyle {\mathcal {Z}}(a^{n}-b^{n})\subset \{1,2,6\}} , and Carmichael's theorem says that the Zsigmondy set of the Fibonacci sequence is { 1 , 2 , 6 , 12 } {\displaystyle \{1,2,6,12\}} , and that of the Pell sequence is { 1 } {\displaystyle \{1\}} . In 2001 Bilu, Hanrot, and Voutier proved that in general, if ( a n ) n 1 {\displaystyle (a_{n})_{n\geq 1}} is a Lucas sequence or a Lehmer sequence, then Z ( a n ) { 1 n 30 } {\displaystyle {\mathcal {Z}}(a_{n})\subseteq \{1\leq n\leq 30\}} (see OEISA285314, there are only 13 such n {\displaystyle n} s, namely 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 18, 30). Lucas and Lehmer sequences are examples of divisibility sequences.

It is also known that if ( W n ) n 1 {\displaystyle (W_{n})_{n\geq 1}} is an elliptic divisibility sequence, then its Zsigmondy set Z ( W n ) {\displaystyle {\mathcal {Z}}(W_{n})} is finite. However, the result is ineffective in the sense that the proof does not give an explicit upper bound for the largest element in Z ( W n ) {\displaystyle {\mathcal {Z}}(W_{n})} , although it is possible to give an effective upper bound for the number of elements in Z ( W n ) {\displaystyle {\mathcal {Z}}(W_{n})} .

See also

References

  1. A. S. Bang (1886). "Taltheoretiske Undersøgelser". Tidsskrift for Mathematik. 5. 4. Mathematica Scandinavica: 70–80. JSTOR 24539988. And Bang, A. S. (1886). "Taltheoretiske Undersøgelser (continued, see p. 80)". Tidsskrift for Mathematik. 4: 130–137. JSTOR 24540006.
  2. Montgomery, H. "Divisibility of Mersenne Numbers." 17 Sep 2001.
  3. Artin, Emil (August 1955). "The Orders of the Linear Groups". Comm. Pure Appl. Math. 8 (3): 355–365. doi:10.1002/cpa.3160080302.
  4. Y. Bilu, G. Hanrot, P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75-122
  5. J.H. Silverman, Wieferich's criterion and the abc-conjecture, J. Number Theory 30 (1988), 226-237
  6. P. Ingram, J.H. Silverman, Uniform estimates for primitive divisors in elliptic divisibility sequences, Number theory, Analysis and Geometry, Springer-Verlag, 2010, 233-263.

External links

Category: