000RM.dvi

408 Digit problems

13.5 The problem of 4n’s

1=nn++nn

2=nn+nn

3=n+nn+n

4=.nn−+..nn

5=.nn++..nn

6=

(n+n+n

n

)

!

7=n−..nn−.n

8=.nn−nn

9=.nn·nn

10 =.nn+nn

11 =.nn +nn

12 =n+.n.n+.n

13 =.nn +

√n

.n

14 =

⌊√

n(n+n)

.n

⌋

15 =.nn+

(√n

.n

)

!

16 =.nn +

(√n

.n

)

!

17 =n+n.n−.n

18 =.nn+.nn

19 =.nn +.nn

20 =.nn +.nn

21 =n+n.n+.n

..

.

Theorem 13.1(Hoggatt and Moser).Letnbe any positive number dif-
ferent from 1 and letpbe any integer greater than 3. Every integer may
be expressed by usingpn’s and a finite number of operator symbols used
in high school texts.

Proof.It is easily verified that for every positive integerk,

loglog√nnlogq√

···√n

n=k,

logn+nnlogq√···√nn=k.

where the base of the second logarithms containsksquare root signs.
These settle the cases of 4 and 5 numbers. For higher values ofp, add an
appropriate numbers of(n−n)+···+(n−n).