Παρασκευή, 23 Οκτωβρίου 2009

24.

  • 24 is the factorial of 4 and a composite number, being the first number of the form 23q, where q is an odd prime.
  • It is the smallest number with exactly eight divisors: 1, 2, 3, 4, 6, 8, 12, and 24. It is a highly composite number, having more divisors than any smaller number. Adding up all the proper divisors of 24 except 4 and 8 gives 24, hence 24 is a semiperfect number.
  • Subtracting one from any of its divisors (except 1 and 2, but including itself) yields a prime number. 24 is the largest number with this property, for to have this property a number cannot be divisible by a prime greater than three, nor can it be divisible by 9 or 16.
  • 24 has an aliquot sum of 36 and the aliquot sequence (24,36,55,17,1,0). Twenty-four is only the aliquot sum of one number the square 529.
  • There are 10 solutions to the equation φ(x) = 24, namely 35, 39, 45, 52, 56, 70, 72, 78, 84 and 90. This is more than any integer below 24, making 24 a highly totient number.
  • 24 is a nonagonal number. This number is also the sum of a twin prime (11 + 13). It is a Harshad number and a semi-meandric number.
  • Together with the number one, 24 is one of the few numbers n for which the sum of μ(d)d2 over the divisors d of n is equal to itself.
  • The product of any four consecutive numbers is divisible by 24. This is because, among any four consecutive numbers, there must be two even numbers, one of which is a multiple of four, and there must be a multiple of three.
  • In 24 dimensions there are 24 even positive definite unimodular lattices, called the Niemeier lattices. One of these is the exceptional Leech lattice which has many surprising properties; due to its existence, the answers to many problems such as the kissing number problem and sphere packing are known in 24 dimensions but not in many lower dimensions. The Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S(5,8,24) and the Mathieu group M24. One construction of the Leech lattice is possible because of the remarkable fact that 12+22+32+...+242 =702 is a perfect square; 24 is the only integer greater than 1 with this property. These properties of 24 are related to the fact that the number 24 also appears in several places in the theory of modular forms; for example, the modular discriminant is the 24th power of the Dedekind eta function.
  • The Barnes-Wall lattice contains 24 lattices.
  • 24 is the highest number n with the property that every element of the group of units (Z/nZ)* of the commutative ring Z/nZ, apart from the identity element, has order 2; thus the multiplicative group (Z/24Z)* = {1,5,7,11,13,17,19,23} is isomorphic to the additive group (Z/2Z)3. This fact plays a role in monstrous moonshine.
  • The 24-cell, with 24 octahedral cells and 24 vertices, is a self-dual convex regular 4-polytope; it has no good 3-dimensional analogue.
  • For any prime p greater than 3, p2 − 1 is divisible by 24 with no remainder.

5 σχόλια:

  1. Μεγαλώνουμε;
    Θα θέλαμε παρόμοια ανάλυση και γι' άλλα νουμεράκια.
    Δεν ξέρουμε αν μας αρέσατε αγαπητέ διότι, ως πλεονέκτης και ποζεράς που είσαστε, χρησιμοποιήσατε όλες τις λέξεις που δεν ξέρουμε και μαθηματικά που επίσης δεν ξέρουμε. :ρ

    ΑπάντησηΔιαγραφή
  2. Εκτός αν έχω απέναντί μου κάποια αξιομνημόνευτη περίπτωση οξυδέρκειας, οι χαρακτηρισμοί "πλεονέκτης και ποζεράς" προδίδουν εμπειρική διαπίστωση και άρα θα ξέρετε αν σας αρέσω!

    Ποια νουμεράκια θα θέλανε εξυπηρέτηση;

    ΑπάντησηΔιαγραφή
  3. Θα προτιμήσω την αξιομνημόνευτη περίπτωση οξυδέρκειας διότι με συμφέρει θαυμάσια. Δεν είναι αλήθεια, αλλά με συμφέρει θαυμάσια.

    Όσο για τα νουμεράκια, θα θέλαμε να μας πείτε για το 30. Είναι η ηλικία που τρέμω.

    ΑπάντησηΔιαγραφή
  4. 30 is the sum of the first four squares, which makes it a square pyramidal number.

    It is a primorial and is the smallest Giuga number.

    30 is the smallest sphenic number, and the smallest of the form 2\cdot 3\cdot r where r is a prime greater than 3. 30 has an aliquot sum of 42; the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises (30,42,54,66,90,144,259,45,33,15,9,4,3,1,0)

    Thirty has but one number for which it is the aliquot sum the square number 841.

    Adding up some subsets of its divisors (e.g., 5, 10 and 15) gives 30, hence 30 is a semiperfect number.

    30 is the largest number such that coprimes smaller than itself are also prime.

    A polygon with thirty sides is called a tricontagon. Both an icosahedron and a dodecahedron have 30 edges.

    E8 has Coxeter number 30.

    30 is a Harshad number.

    Since any group G such that |G| = pnm, where p does not divide m, has a subgroup of order pn, and 30 is the only number less than 60 that is not either a prime or of the above form, it is the only candidate for the order of a simple group less than 60 that one needs other methods to reject.

    The Saros number of the solar eclipse series which began on October 2, 2069 BC and ended on March 18, 572 BC. The duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series which began on June 19, 1803 BC and ended on August 18, 487 BC. The duration of Saros series 30 was 1316.2 years, and it contained 74 lunar eclipses.

    The number of days in the months April, June, September and November (and in unusual circumstances February - see February 30)

    The total number of major and minor keys in Western tonal music, including enharmonic equivalents

    The code for international direct dial phone calls to Greece

    The number of tracks on The Beatles' eponymous album, usually known as The White Album

    Judas Iscariot betrayed Jesus for 30 pieces of silver. Matthew 26:15.

    Slang for pornography due to its representation as Roman numeral XXX.

    At age 30 (according to most biblical scholars) Jesus of Nazareth was baptized by John the Baptist, at the beginning of his public ministry of teaching and healing.

    One of the rallying-cries of the 1960s student/youth protest movement was the slogan, 'Don't trust anyone over thirty'.

    In 'The Myth of Sisyphus' the French existentialist Albert Camus comments that the age of thirty is a crucial period in the life of a man, for at that age he gains a new awareness of the meaning of time.

    In Franz Kafka's novel 'The Trial' Joseph wakes up on the morning of his thirtieth birthday to find himself under arrest for an unspecified crime.

    The number of uprights that formed the Sarsen Circle at Stonehenge, also the supposed number of holes forming the arrays of Y and Z Holes at Stonehenge.

    The number of variations in Bach's Goldberg Variations

    30 BC:

    Death of Mark Antony, Roman politician and general.

    Death of Pharaoh Cleopatra VII of Egypt.

    First possible date for the invention of the wheelbarrow in history.

    30 AD:

    Death of Judas Iscariot, disciple of Jesus, reportedly committed suicide shortly before the death of his teacher.

    Death of Jesus (born about 4 BC)—most widely accepted date by modern scholars.

    ΑπάντησηΔιαγραφή
  5. !!!
    Μήπως είσαι τζίνι;
    Μας κακομαθαίνετε, αγαπητέ συγγραφέα.
    Μη το κάνετε, διοτί μετά θα ζητάμε κι άλλο.
    Ευχαριστώ!

    ΑπάντησηΔιαγραφή