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sci.math FAQ: Unsolved Problems


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Archive-Name: sci-math-faq/unsolvedproblems
Last-modified: 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    SI--Prefix
-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   (2n-1)-illion n-illion
6*n+3 (2n)-illion   n-illiard
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?






     _________________________________________________________________



    alopez-o@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.




     _________________________________________________________________




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