ARITHMETIC OF PENTAGONAL NUMBERS
RODNEY T. HANSEN
Montana State University, Bozeman, Montana
The pentagonal numbers are the integers
Pn = f
On - 1),
n = 1,2,—
.
Each number p can also be derived by summing the first n terms of the
arithmetic progression
1, 4, 7, 10, 13, • •• , 3m - 2 .
Geometrically, considering regular pentagons nomothetic with respect to one
of the vertices and containing 2, 3, 4, • • • , n points at equal distances along
each side, the sum of all points for a given n yields p . Pictorially we have
the following: [1, p. 10]
In this paper we shall give several algebraic identities involving pentagonal
numbers of different orders. The principal result is that an infinite number
of pentagonal numbers exist which a r e , at the same time, the sum and difference of distinct pentagonal numbers. A similar result for triangular numbers
has been found by W. Sierpinski [ 2 , pp9 31-32].
83
84
ARITHMETIC O F PENTAGONAL NUMBERS
[Feb.
A table of p f s will f i r s t be c o n s t r u c t e d .
Pn
0
1
2
3
4
5
6
7
8
9
0
2
1
3
4
_
j -5" ~TF"
22"
35~
330
176
287
145
210
247
925
852
590
651
715
782
1,335
1,426
1,717 1,820
1,520 1,617
2,380 2,501 2,625 2,752
2,882 3,015
3,725
3,876 4,030 4,187 4,347 4,510
5,370 5,551 5,735 5,922 6,112 6,305
7,315
7,526
7,740
7,957
8,177 8,400
9,560
9,801 10,045 10,292 10,542 10,795
12,105 12,376 12,650 12,927 1 3 , 2 0 7 . 13,490
7
6
5
—
9
8
70" ~
92"
ITT
532
376
425
477
1,001 1,080 1,162 1,247
1,926 2,035 3,147 2,262
3,115 3,290 3,432 3,577
4,676 4,845 5,017 5,192
6,501 6,700 6,902 7,107
8,626 8,855 9,087 9,322
11,051 11,310 11,572 11,837
13,776 14,065 14,357 14,652
~wr
We note from the above-given a r i t h m e t i c p r o g r e s s i o n that
p - p - = 3n - 2 ,
*n
*n-l
for
n = 2, 3 , • • • ,
' '
and from the above table that
P8 = P4
+
P7»
P24 == P?
+
P23>
P49
=
PiO + P48>
and
P83 = Pl3
+
P82 •
Noting that the f i r s t t e r m on the r i g h t of e a c h of the above equalities i s of the
form
3n + 1, we find that
?3nHhl
=
• ^ 2 L 1 L [3(n
+
1) - 1 ] = I ( 2 7 n 2
+
15n
Setting
p
- pF
^m
m - 11
=
3 m - 2 = ~
(27n2 + 15n + 2)
2N
we have
m = ~ (9n2 + 5n + 2 ) ,
an i n t e g e r .
The f i r s t t h e o r e m follows.
+
2
)
1970]
ARITHMETIC O F PENTAGONAL NUMBERS
T h e o r e m 1.
85
F o r any i n t e g e r n > 1,
P
i(9n 2 +5n-f-2)
~
P
<3n+1)
P
#(9n+5)
A s u b s e t of the above defined pentagonal n u m b e r s yields o u r m a i n r e s u l t .
T h e o r e m 2.
F o r any positive i n t e g e r n ,
1 [9(3n) 2 +5(3n)+2]
[3(3n)+l]
3n j - 9 ( 3 n ) + 5 ]
Pi
~P
i-(6561n 4 +2430n 3 +4B5n 2 +5Gn+8)
~(6561n 3 +2430n 2 +495n+50)
Proof.
F i r s t it i s n e c e s s a r y to e x p r e s s
Pi
•i(81n 2 +15n+2)
in t e r m s of n.
i ( 8 1 n 2 + 15n + 2) ( r i
-,
=
i
3 A(81n 2 + 15n + 2) Pl
J
i(81n 2 +15n+2)
*
M~ ( 1 9 , 6 8 3 n 4 + 7290n 3 + 1485n 2 + 150n + 8)
Equating p g - pQ_±
.
to
P-s
^(81n 2 +15n+2)
yields
s = i (6561n 4 + 2430n 3 + 495n 2 + 50n + 8)
o
By m a t h e m a t i c a l induction on n we have that s i s an integer; completing the
proof.
86
ARITHMETIC O F PENTAGONAL NUMBERS
[Feb.
F o r n = 1 and 2 we h a v e s for e x a m p l e ,
=
P49
PiO
+
P48
=
Pii93 " P1192
or
3577 = 145 + 3432 = 2,134,277 - 2,130,700
and
Pi78 -
Pi9
+
Pl77 ~ Pi5,813 " Pi5,812
or
47,437 = 532 + 46,905 = 375,068,547 - 375,021,110 .
A r a t h e r c u r i o u s r e l a t i o n s h i p e x i s t s between n and p
; n a m e l y , that
each positive i n t e g e r c a n b e e x p r e s s e d in an infinite n u m b e r of ways a s a quadr a t i c e x p r e s s i o n involving a pentagonal n u m b e r .
T h e o r e m 3.
Any positive i n t e g e r n can be e x p r e s s e d a s
1 + -J 1 + 24p
*
^s-n
6-s
for any positive i n t e g e r
Proof.
s.
F r o m the definition of a pentagonal n u m b e r we have
p
*sn
= » (3 • sn - 1) =
2
3(sn)2
9
2
~
sn
0 = 3s 2 n 2 - sn - 2p
^sn
+s ± ^ ( - s ) 2 - T ( 3 s 2 ) ( - 2 p s n )
1 ± ^ 1 + 24p"
sn
_
*
*sn
6s
2 • 3s 2
1970]
ARITHMETIC O F PENTAGONAL NUMBERS
87
Taking the positive r o o t , the d e s i r e d r e s u l t i s obtained.
The pentagonal n u m b e r s a r e not closed with r e s p e c t to the o p e r a t i o n of
multiplication.
However* the following t h r e e c a s e s a r e quickly verified. 9
P8T = P2 P39» Pi8T
=
and
P4 P40*
P392 = P? P47 •
It i s not known if an infinite n u m b e r of such p a i r s exist.
REFERENCES
1.
J . V. Uspensky and M. A. H e a s l e t , E l e m e n t a r y N u m b e r T h e o r y , M c G r a w Hill, New York, 1939.
2.
W. S i e r p i n s k i , "Un t h e o r e m e s u r l e s n o m b r e s t r i a n g u l a i r e s , f f
Elemente
P e r M a t h e m a t i k , Bank 2 3 , N r 2 ( M a r z , 1968), pp. 31-32.
• • * •
*
CORRECTIONS
P l e a s e m a k e the following changes in " A s s o c i a t e d Additive D e c i m a l Digital
B r a c e l e t s , " a p p e a r i n g in the Fibonacci Q u a r t e r l y in O c t o b e r , 1969:
On page 288, line 2 5 , change " t e r m s " to "forms* "
On page 289, line 2 , change " 8 " to r e a d " B . "
On page 290, line 1 1 , change " 7 8 4 2 " to "6842. "
On page 290, line 1 3 , change "and" to r e a d "And, "
On page 294, line 2 0 , change "19672" to r e a d "1967). "
On page 294, line 2 6 , change "1969" to r e a d "1959. "
P l e a s e change the f o r m u l a s given in "Diagonal Sums of G e n e r a l i z e d
P a s c a l T r i a n g l e s , " page 3 5 3 , Volume 7, No. 5 , D e c e m b e r , 1969, l i n e s 11 and
1 2 , to r e a d