(R), green (G), and blue (B), they can be grouped in five different ways: For example, if there are three balls, colored red (such as differently colored balls) can be grouped into sets (such as buckets) A number n is called tri-automorphic if 3 n 2 ends in n for example 667 is tri-automorphic because 3 ×Ī Bell number is the number of ways that n distinguishable objects Not all trimorphic numbers are automorphic. A number n is called trimorphic if n 3 ends in n. For instance 5 is automorphic, becauseĥ 2 = 25, which ends in 5. The system actually derives from Hindu mathematics.Īn automorphic number, also known as an automorph, is a number n whose square ends in n. An unusually high proportion of the numbers in amicable pairs endsĪrabic numerals are numerals written with Arabic digitsĪlone: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or in combination: 10, 11, 12. No amicable pair is known in which one of the two numbers is a Today, the tally of known amicable numbers has grown to about two and half This second lowest pair of all had been completely overlooked. Startled the mathematical world by announcing that the numbers 1184 andġ210 were friendly. Nicoló Paganini (not the violinist!), a 16-year-old Italian, In the 18th century Leonhard Euler drew up a list of more than 60. In 1636 Pierre de Fermat rediscovered the amicable pair 1726 two years later René Descartes rediscovered a third pair, 93635056. This pair was known to the ancient Greeks,Īnd the Arabs found several more. (An aliquot part is any divisor that doesn't include the It isn't known whether there is an odd number n whose divisors (excluding itself) sum to n – 1.Īmicable numbers are pairs of numbers, also known as friendly numbers, Of 2 is a deficient number (one that is less than the sum of its properĭivisors), but only just. Numbers is countably infinite, the set of transcendental numbers is uncountablyĪn almost perfect number is a description sometimes applied to the powers of 2 because If a real number is not algebraic,Īlmost all real numbers are transcendental because, whereas the set of algebraic The square root of two isĪn algebraic number of degree two because it is a root of the quadratic equation x 2 – 2 = 0. Of degree one, because it is a root of the linear equation bx – a = 0. Where a and b are non-zero integers, is an algebraic number Published posthumously in 1713 by his nephew Nikolaus Bernoulli.Īn algebraic number is a real number that is Looking like an "8" on its side, was introduced in 1655 by John Wallis in his Arithmetica infinitorum but didn't appear in print until the Ars conjectandi by Jakob Bernoulli, An earlier (and still used) symbol for infinity, The various orders, or sizes, of infinity:Īleph-null, aleph-one, etc. Any divisor of a deficient (or perfect) number is deficient.Ī number that is not abundant or deficient is known as a perfectĪleph is the first letter of the Hebrew alphabet. The first few deficient numbers are 1, 2, 3, 4, 5,ħ, 8, and 9. It isn't known if there are any odd weird numbers.Ī deficient number is one that is greater than the sum Than n, but n is not equal to the sum of any subset of itsĭivisors. In other words, n is weird if the sum of its divisors is greater The sum of its aliquot parts is 1 + 2 + 3 + 4 + 6 = 16 – followedĪ weird number is an abundant number that is not semiperfect An abundant is a number that is smaller than the sum
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