Martin Gardner’s Lucky Number
Hunting for prime numbers, those evenly divisible only by
themselves and 1, requires a sieve to separate them from the rest.
For example, the sieve of Eratosthenes, named for a Greek
mathematician of the third century B.C., generates a list of prime
numbers by the process of elimination.
To find all prime numbers less than, say, 100, the hunter writes
down all the integers from 2 to 100 in order (1 doesn’t count as a
prime). First, 2 is circled, and all multiples of 2 (4, 6, 8, and so on)
are struck from the list. That eliminates composite numbers that
have 2 as a factor. The next unmarked number is 3. That number is
circled, and all multiples of 3 are crossed out. The number 4 has
already been crossed out, and its multiples have also been
eliminated. Five is the next unmarked integer. The procedure
continues in this way until only prime numbers are left on the list.
Though the sieving process is slow and tedious, it can be continued
to infinity to identify every prime number.
Other types of sieves isolate different sequences of numbers.
Around 1955, the mathematician Stanislaw Ulam (1909-1984)
identified a particular sequence made up of what he called "lucky
numbers," and mathematicians have been playing with them ever
since.
Starting with a list of integers, including 1, the first step is to cross
out every second number: 2, 4, 6, 8, and so on, leaving only the odd
integers. The second integer not crossed out is 3. Cross out every
third number not yet eliminated. This gets rid of 5, 11, 17, 23, and
so on. The third surviving number from the left is 7; cross out every
seventh integer not yet eliminated: 19, 39 ... Now, the fourth
number from the beginning is 9. Cross out every ninth number not
yet eliminated, starting with 27.
This particular sieving process yields certain numbers that
permanently escape getting killed. That’s why Ulam called them
"lucky."
What’s remarkable is that the "luckies," though generated by a sieve
based entirely on a number’s position in an ordered list, share
many properties with primes. For example, there are 25 primes less
than 100 and 23 luckies less than 100. Indeed, it turns out that
primes and luckies come up about equally often within given ranges
of integers. The distances between successive primes and the
distances between successive luckies also keep increasing as the
numbers increase. In addition, the number of twin primes – primes that
differ by 2 – is close to the number of twin luckies.
Perhaps the most famous unsolved problem involving primes is the
Goldbach conjecture, which states that every even number greater than
2 is the sum of two primes. Luckies are featured in a similar conjecture,
also unsolved: Every even number is the sum of two luckies. Computer
searches have reached at least 100,000 without finding an exception.
Martin Gardner describes many more features of lucky numbers in a
delightful article in a recent issue of The Mathematical Intelligencer.
"There is a classic proof by Euclid that there is an infinity of primes," he
writes. "Although it is easy to show there is also an infinity of lucky
numbers, the question of whether an infinite number of luckies are
primes remains, as far as I know, unproved."
How did the topic of lucky numbers happen to come up? The house
where Gardner grew up in Tulsa, Okla., had the address 2187 S.
Owasso. "Of course I never forgot this number," he says. It also
happens to be one of the lucky numbers. Gardner’s imaginary friend,
the noted numerologist Dr. Irving Joshua Matrix, can readily find
additional remarkable properties associated with that number.
Exchange the last two digits of 2187 to make 2178, multiply by 4, and
you get 8712, the second number backwards. Take 2187 from 9999
and the result is 7812, the number in reverse. Moreover, the first four
digits of the constant e, 2718, and the number of cubic inches in a
cubic foot, 12^3 = 1728, are each permutations of 2187!
However, to those inclined to seeing meaning in certain numbers, Dr.
Matrix issues the following warning: "Every number has endless
unusual properties."
Copyright © 1997 by Ivars Peterson.
Martin Gardner’s Lucky Number
4. Find seven phrasal verbs in the text. Give the French for the
verbs, both alone and with the particle
1. What is a lucky number?
a) in everyday life
English
Translation without
particle
Translation with
particle
b) in mathematics, according to this text
2. Re-read paragraph 5, and work out whether the following
numbers are ‘lucky’. You may need scrap paper.
NUMBER
9
17
43
49
67
91
99
LUCKY
UNLUCKY
5. Form questions related to the underlined elements :
a) Martin Gardner’s Lucky number is 2187.
b) Around 1955 Ulam discovered lucky numbers.
3. Find words equivalent to :
search
manner
boring
eternally
qualities
theory
characteristics
edition
subject
easily
prone
strange
c) Eratosthenes was born in Greece.
sift
remaining
produce
surprising
inside
attained
pleasant
if
famous
furthermore
alert
d) There is an infinity of primes.
e) Euclid proved that primes were infinite.
f)
Ulam has been dead for 16 years.
g) The house where Gardner grew up had the address 2187.
h) Gardner wrote a column for Scientific American every month.
Martin Gardner’s Lucky Number
6. Complete with quantifiers from the list :
all, any, every, many, much, no, several, some, two
__________ mathematician has heard of it, and __________ people have
tried to prove the Goldbach conjecture, but so far __________-one has
succeeded. It states that __________ even integer is the sum of
__________ primes. Prime numbers and Lucky numbers share
__________ common properties, but not __________ of them have been
proved. __________ primes, known as Mersenne primes are related to
perfect numbers, and __________ has been written about these.
7. What is a perfect number?
Martin Gardner’s Lucky Number
5.
a. Whose lucky number is 2817?
Answers :
1. A number that
b. When did Ulam discover lucky numbers?
a) brings good luck
c. Where was Eratosthenes born?
b) is not eliminated by the iterative process described above.
d.
2. list of luckies : 1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73
75 79 87 93 99 105 111 115 127 129 133 135 141 151 159 163
169 171 189 193 195
3. search (hunt), sift (sieve), manner (way), remaining (left), boring
(tedious), produce (yield), eternally (permanently) surprising
(remarkable) qualities (properties), inside (within), theory
(conjecture), attained (reached), characteristics (features),
pleasant (delightful), edition (issue), if (whether), subject (topic),
famous (noted), easily (readily), furthermore (moreover), prone
(inclined), alert (warning), strange (unusual)
How many primes are there?
e. Who proved that primes were infinite?
f.
For how long has Ulam been dead?
g. Which house had the address 2187?
h. How often did Garner write a column for Scientific American?
6. Every mathematician has heard of it, and many people have tried to
prove the Goldbach conjecture, but so far no-one has succeeded. It
4.
states that any even integer is the sum of two primes. Prime numbers
English
Translation without
particle
Translation with
particle
and Lucky numbers share several common properties, but not all of
write down
Écrire
Ecrire, noter
them have been proved. Some primes, known as Mersenne primes are
strike from
Frapper
Eliminer
related to perfect numbers, and much has been written about these.
cross out
Traverser
Barrer
make up
Faire
Maquiller, inventer
turn out,
Tourner
S’avérer
come up
Venir
Se produire
grow up
Croître
Grandir
7. Equal to the sum of its divisors (excluding itself) 6, 28, 496, 8128
All equal
All triangular
Binaries follow a pattern : 110, 11100, 1111000, 111110000