In the early part of the 20th Century, Robert Daniel Carmichael studied integers that were pseudoprimes to the base of every positive integer.  Such pseudoprimes became known as Carmichael numbers.  A Carmichael number is an odd composite number n which satisfies b^n-1  1 (mod n) for all positive integers b with (b, n) = 1.

Carmichael numbers are rare.  There are only 43 Carmichael numbers < 10⁶, 2163 <2.5 × 10¹⁰, 105212 <10¹⁵, and 246683<10¹⁶.  Despite their scarcity, Carmichael conjectured that there is an infinitude of Carmichael numbers in 1912, and Alford, Granville, and Pomerance proved his conjecture in 1994.  Alford, Granville, and Pomerance also determined that for large values of n (n ~10⁷) there are approximately n^2/7 Carmichael numbers.  Mathematicians also refer to Carmichael numbers as absolute pseudoprimes, and they satisfy Korselt's criterion.

Carmichael numbers possess many interesting qualities.  First, all Carmichael numbers have at least three prime factors.  If a Carmichael number has only three prime factors, there are only a finite number of ways to construct it given one of the three factors.  Also, a number n satisfies the Carmichael condition if and only if n = q₁ q₂ ^...q_k where the q_j's are distinct primes that satisfy (q_j - 1) |  (n - 1) for all j.  Further, every Carmichael number is squarefree, and an odd composite squarefree number n is a Carmichael number if and only if n divides the denominator of the Bernoulli number B_n-1.

The first ten Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, and 29341.  Numbers of the form (6k + 1)(12k + 1)(18k + 1) are also Carmichael numbers  if each of the factors (6k + 1, 12k + 1, and 18k + 1) is prime.  Currently, H. Dubner has discovered a Carmichael number with three factors with 10200 digits in it.

Author: Tony Brinsko

References:
Rosen, Kenneth H.  Elementary Number Theory and Its Applications.  Third Edition.  Menlo Park: Addison-Wesley Publishing Company, 1993.

http://www.astro.virginia.edu/~eww6n/math/CarmichaelNumber.html
http://www.astro.virginia.edu/~eww6n/math/CarmichaelCondition.html

In an era in which society believed that the woman's place is in the home, Charlotte Angas Scott emerged as one of England's first women to obtain a doctorate in mathematics.  Despite the disapproval from her colleagues and society, Scott pursued her endeavors to seek equality for women.  Her work and aspirations, however, were not hindered by these challenges.  Scott was successful in her aspirations and is considered to be the pioneer for the advancement of women's role in the fields of mathematics and the sciences.

Charlotte Scott's desire to pursue her education in mathematics was sparked by her parent's guidance and support.  As a young child, her parents provided her with math tutors and a secondary education, which at the time were not available to women.  By obtaining a secondary education, Scott was able to win a scholarship and attend college at the age of 18.   In her fourth year of college, she completed her final examinations and was ranked the top 10 in her graduating class.  However, since receiving a bachelor's degree with honors was awarded exclusively to male students, she was not allowed to attend the awards ceremony.  This hurdle did not discourage her, but only motivated her to work even harder. Earning a Bachelor of Science and doctorate, degrees of the highest scholastic honor enabled her to position herself and act against sexual discrimination.  Her first project resulted in all women being able to take college examinations and having their names announced publicly with the men.  Scott's further accomplishments include administrating admissions requirements at the college she taught, publishing work in mathematical journals and becoming the first woman (later vice-president) on the first Council when the American Mathematical Society began.  Under her guidance, women began to undertake more active roles in the study of mathematics.

Author: Henry Sheen

References:
Grinstein, Louise S.  Women of Mathematics.  New York:  Greenwood Press, 19687.

Kenschaft, Patricia.  "Charlotte Angas Scott, 1858-1931." The College Mathematics Journal 18 (1987), 98-110.