I 60 THE FASCINATION OF NUMBERS

the quadratic n2+ (¢ —1)n+N=o0, we have:

o etk \/(a—l)é‘%fi

M TR ST A RS SRR

 

2
~ —a+1+V(a+x)?
i SR
q _—at1d(a+s)

| 2
14X I1—2a—X

The second alternative root is impossible, which leaves the
value of

I1-+x
g

We now take the number %29 as an example. 4N 41 be-
comes 4(729) +1=2917. If 729 is composite, then we should
be able to find one or more odd values (other than 1) for x so

n

 

that
— 2
2—%}% =integral multiple of 2
In fact, we find that:
2917 —52
511 > =48

1+
B

whence x=5 and a=241. The vaiue of n is then —=
2

S L R R S AR R R A s
T 1 A RRBEEHRBHOABOHAENTBELUBNNRE 3

   
  

The number 729 can therefore be expressed as a series of
consecutive odd numbers of three terms, the first term being

241.

e

2411243 +245=729
It should be noted that other values for x can be found,
according to the number of different factors of 729.
A5 2
For example: N

(17+1)

TIISE TR IY LI

e T

it (T T AN
) TP bl WA i 1t