64 Catalan, Eugène Charles

remainder of 5 + 9 + 4 + 3 + 2 + 6 + 4 + 1 = 34 when

divided by 9, which corresponds to a remainder of 3 +

4 = 7. Any sets of digits that sum to 9, such as the 5

and the 4 in the first and third positions of the number

above, can be ignored when performing this calcula-

tion, for they will not contribute to the remainder.

The method of “casting out nines” is the process of

deleting groups of digits summing to 9. The sum of the

digits that survive is the remainder that number yields

upon division by 9. For example, 59,432,641 →

932,641 →934 →34 shows, again, that this number

leaves a remainder of 3 + 4 = 7 when divided by 9.

This method is often used to check arithmetical

work. For example, we can quickly determine that 563

×128 cannot equal 72,364. Upon division by 9, 563

leaves a remainder of 5, 128 a remainder of 2, and so

their product leaves a remainder of 5 ×2 = 10, which is

1. Yet 72,364 has a remainder of 4.

Long lists of additions, subtractions, and multipli-

cations can be quickly checked this way. Of course,

errors may still be present if, by chance, remainders

happen to match. For example, casting out nines will

not detect that 632 ×723 = 459,636 is incorrect.

Catalan, Eugène Charles (1814–1894) Belgian Num-

ber theory Born on May 30, 1814, mathematician

Eugène Catalan is best remembered for his work in

NUM

-

BER THEORY

and for the famous series of numbers that

bears his name. In 1844 Catalan conjectured that 8 and 9

are the only two consecutive integers that are both non-

trivial powers (8 = 23and 9 = 32). Establishing this claim,

today known as the C

ATALAN CONJECTURE

, stymied

mathematicians for over a century. It was only recently

resolved.

Catalan’s career in academia was turbulent. After

entering the École Polytechnique in 1833 he was

expelled the following year for engaging in radical

political activity. He was later given permission to

return to complete his degree and in 1838 was offered

a post as lecturer at the institution, which he accepted.

His political conduct, however, hampered his ability to

advance beyond this entry-level position.

Catalan worked in the field of

CONTINUED FRAC

-

TION

s and achieved some fame for publishing a sim-

plified solution to L

EONHARD

E

ULER

’s “polygon

division problem.” This challenge asks for the number

of ways to divide a regular polygon into triangles

using nonintersecting diagonals. While not being the

first to solve the problem (in fact the problem was

first stated and solved by 18th-century Hungarian

mathematician J. A. Segner and then studied by

Euler), Catalan used an approach that was particu-

larly elegant. His 1838 paper on the topic, “Note sur

une équation aux differences finie” (Note on a finite

difference equation), was very influential because of

the method it detailed. The sequence of numbers that

arise in the study of the problem are today called the

C

ATALAN NUMBERS

. They remain his standing legacy.

Catalan died on February 14, 1894, in Liège,

Belgium.

Catalan conjecture In his 1844 letter to Crelle’s

Journal, E

UGÈNE

C

HARLES

C

ATALAN

conjectured that

the integers 8 and 9 are the only two consecutive inte-

gers that are both powers (8 = 23and 9 = 32). He was

not able to prove his claim, and establishing the truth

or falsehood of the conjecture became a longstanding

open problem. In April 2002, amateur mathematician

Preda Mihailescu announced to the mathematics com-

munity that he had completed a proof demonstrating

Catalan’s assertion to be true. Beforehand, Mihailescu

had proved a series of related results, all while working

at a Swiss fingerprinting company and exploring math-

ematics as an outside interest. Mathematicians are cur-

rently reviewing his final step of the work.

Catalan numbers In 1838 E

UGÈNE

C

HARLES

C

ATA

-

LAN

studied the problem of finding the number of dif-

ferent ways of arranging npairs of parentheses. For

example, there is one way to arrange one set: ( ), two

ways to arrange two pairs: ( ) ( ) and (( )), and five

ways to arrange three pairs: ((( ))), (( )( )), (( )) ( ), ( )

(( )), and ( ) ( ) ( ). Is there a general formula for the

number of ways to arrange npairs of parentheses? This

puzzle is today known as “Catalan’s problem.” As

Catalan showed, the solution is given by the formula:

yielding the sequence of numbers C1= 1, C2= 2, C3=5,

C4= 14, C5= 42, …, now called the Catalan numbers. It

is convenient to set C0= 1. Some algebraic manipulation