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From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)
Subject: sci.math FAQ: Unsolved Problems
Summary: Part 18 of many, New version,
Originator: alopez-o@neumann.uwaterloo.ca
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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.
_________________________________________________________________