See reader questions & answers on this topic!  Help others by sharing your knowledge ArchiveName: scimathfaq/unsolvedproblems Lastmodified: December 8, 1994 Version: 6.2 NAMES OF LARGE NUMBERS & UNSOLVED PROBLEMS _________________________________________________________________ * Names of large numbers * Does there exist a number that is perfect and odd? * Collatz Problem * Goldbach's conjecture * Twin primes conjecture _________________________________________________________________ Names of large numbers Naming for 10**k: k American European SIPrefix 24 Yocto 21 Zepto 18 QUINTILLIONTH Atto 15 QUADRILLIONTH Femto 12 TRILLIONTH Pico 9 BILLIONTH Nano 6 MILLIONTH Micro 3 THOUSANDTH Milli 2 HUNDREDTH Centi 1 TENTH Deci 1 TEN Deca 2 HUNDRED Hecto 3 THOUSAND Kilo 4 Myria (?) 6 Million Million Mega 9 Billion Milliard Giga In italy (Thousand Milliards) 12 Trillion Billion Tera 15 Quadrillion Billiard Peta 18 Quintillion Trillion Exa 21 Sextillion Trilliard Zetta 24 Septillion Quadrillion Yotta 27 Octillion Quadrilliard 30 Nonillion Quintillion (Noventillion) 33 Decillion Quintilliard 36 UNDECILLION Sextillion 39 DUODECILLION Sextilliard 42 tredecillion Septillion 45 quattuordecillion Septilliard 48 quindecillion Octillion 51 sexdecillion Octilliard 54 septendecillion Nonillion (Noventillion) 57 octodecillion Nonilliard (Noventilliard) 60 novemdecillion Decillion 63 VIGINTILLION Decilliard 6*n (2n1)illion nillion 6*n+3 (2n)illion nilliard 100 Googol Googol 303 CENTILLION 600 CENTILLION 10^100 Googolplex Googolplex The American system is used in: US, ... The European system is used in: Austria, Belgium, Chile, Germany, the Netherlands, Italy (see excepcion) hv@cix.compulink.co.uk (Hugo van der Sanden): To the best of my knowledge, the House of Commons decided to adopt the US definition of billion quite a while ago  around 1970?  since which it has been official government policy. dik@cwi.nl (Dik T. Winter): The interesting thing about all this is that originally the French used billion to indicate 10^9, while much of the remainder of Europe used billion to indicate 10^12. I think the Americans have their usage from the French. And the French switched to common European usage in 1948. gonzo@ing.puc.cl (Gonzalo Diethelm): Other countries (such as Chile, my own, and I think most of Latin America) use billion to mean 10^12, trillion to mean 10^18, etc. What is the usage distribution over the world population, anyway? _________________________________________________________________ alopezo@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 Does there exist a number that is perfect and odd? A given number is perfect if it is equal to the sum of all its proper divisors. This question was first posed by Euclid in ancient Greece. This question is still open. Euler proved that if N is an odd perfect number, then in the prime power decomposition of N , exactly one exponent is congruent to 1 mod 4 and all the other exponents are even. Furthermore, the prime occurring to an odd power must itself be congruent to 1 mod 4. A sketch of the proof appears in Exercise 87, page 203 of Underwood Dudley's Elementary Number Theory, 2nd ed. It has been shown that there are no odd perfect numbers < 10^(300) . _________________________________________________________________ Collatz Problem Take any natural number m > 0 . n : = m; repeat if ( n is odd) then n : = 3*n + 1 ; else n : = n/2 ; until ( n = = 1 ) Conjecture. For all positive integers m, the program above terminates. The conjecture has been verified up to 7 * 10^(11) . References Unsolved Problems in Number Theory. Richard K Guy. Springer, Problem E16. _________________________________________________________________ Does there exist a number that is perfect and odd? A given number is perfect if it is equal to the sum of all its proper divisors. This question was first posed by Euclid in ancient Greece. This question is still open. Euler proved that if N is an odd perfect number, then in the prime power decomposition of N , exactly one exponent is congruent to 1 mod 4 and all the other exponents are even. Furthermore, the prime occurring to an odd power must itself be congruent to 1 mod 4. A sketch of the proof appears in Exercise 87, page 203 of Underwood Dudley's Elementary Number Theory. It has been shown that there are no odd perfect numbers < 10^(300) . _________________________________________________________________ Collatz Problem Take any natural number m > 0 . n : = m; repeat if ( n is odd) then n : = 3*n + 1 ; else n : = n/2 ; until ( n = = 1 ) The conjecture has been verified for all numbers up to 7 * 10^(11) . References Unsolved Problems in Number Theory. Richard K Guy. Springer, Problem E16. Elementary Number Theory. Underwood Dudley. 2nd ed. _________________________________________________________________ Goldbach's conjecture This conjecture claims that every even integer bigger equal to 4 is expressible as the sum of two positive prime numbers. It has been tested for all values up to 2*10^(10) . _________________________________________________________________ Twin primes conjecture There exist an infinite number of positive integers p with p and p + 2 both prime. See the largest known twin prime section. There are some results on the estimated density of twin primes. _________________________________________________________________ User Contributions:
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