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Note on d^n (mod q) and an amusing result for trigonometricfunctionsCitation for published version (APA):Rienstra, S. W. (1986). Note on d^n (mod q) and an amusing result for trigonometric functions. (WD report; Vol.8605). Nijmegen: Radboud Universiteit Nijmegen.
Document status and date:Published: 01/01/1986
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Download date: 07. Jun. 2020
report WD 86-05
Note on n
d (mod q) and an amusing result for trigonometric functions
S.W. Rienstra
December 1986
Wiskundige Dienstverlening Faculteit der Wiskunde en Natuurwetenschappen Katholieke Universiteit
Toernooiveld
6525 ED Nijmegen
The Netherlands
Note on dn (mod q) and an amusing result
for trigonometric functions
S. W. Rienstra
Mathematics Consulting Department
Katholieke Universiteit Nijmegen, The Netherlands
December 1986
Introduction
While playing with my pocket calculator some time ago, I found an amusing property of trigonometric
functions in degree mode . It amounts to the constancy of 10n (mod 360) for n ~3, and some related
results. These results and its generalisations are the subject of this note . Although similar relations for
other moduli are well known (for example, for 2,3,5,7,9,10,11,13, ... ; see [1], (9.5)), the one for 360, in
spite of its applicability to trigonometric functions, seems to have escaped attention, or, at least, seems
to have not yet been published. The same appears to be true for the generalisation to be presented here
(cf. [2]). I have published the present result partially, and in a very condensed form, as a puzzle in [3],
but by this note I would like to give it a wider audience.
Special case
If .n ~3, then
= 1000 + 360-fold, since 9000 = 25x360.
So
wn = 1000 = -80(mod 360) if n~3 ,
making
sin(lon) = -sin(80).
Similarly,
10n = 100(mod 180) if n~2,
making
tan(1on) = tan(100).
An immediate consequence is then:
consider a natural number x in decimal representation n
x = ~ ailOi i=O
then n
x = HP~ ai + 100a2 + 10a1 + a0 (mod 360). i=3
Obviously, the term 103~ai can be reduced further and further, down to -80 times a one-digit
number, in this way greatly simplifying the evaluation of sin(x) (or cos(x), or similarly tan(x) ) when x
is a large number.
-- 1 --
General case
The above result can easily be generalised, and we investigate for which conditions dn is constant
(mod q) in n ";3k, for integers d, q ";31, and k ";30.
Theorem: dn = dk(mod q) for all n";3k if and only if ql(d-1)dk.
Proof: d=O and d=1 are trivial. If d*O,=;q :
(i). if dn = dk (mod q) for n ";3k, then in particular for n =k + 1
dk+ 1-dk = (d-1)dk = Nq for some N #.
(ii).if ql(d-1)dk and n";3k,then
dn-k 1 dn=dk+(d-1)dk - =dk(modq),
d-1
since (d-1)l(dn-k_1) # .
Assume from here on ql(d-1)dk. Let c denote the least non-negative residue of dn(mod q) . If
0 ~ dk < q, then c = dk; otherwise we can construct c as follows. Write d -1 = q IP b dk = q2[J 2
with q = q 1q2 and q"q2 > 0. This decomposition is unique, since (d-1,dk) = 1. Let pz denote the
least non-negative residue of p 2(mod q 1). Then
c = q2[Ji. .
It is not difficult to prove the corollaries, that c =0 if and only if
if ql(d-1) and q*l.
Now consider a number x, written as
x = y + dkx 0
, and c=1 if and only
with y <dk. Starting with x 0, we construct a sequence (x j) via repeated summation of the digits in base
d (where for the moment d is taken positive; this assumption can be relaxed).
no xo = L ao;d; ,
i=O
no nl
x, = L ao; = L a!idi , i=O i=O
Xm = amo ·
(Note that the sequence n0,nb···,nm = 0 decreases exponentially, since nj = [dlog Xj], and
Xj+! ~ (d -1)(nj + 1) ) . Then we have
x = y + dkx 1 = ... = y + cxm (mod q).
-- 2 --
From the general result we can derive specific examples. Of course, for d=10, the cases q=360, k=3,
and q=180, k=2 follow readily, as well as the well-known q=3 and 9 (k=O) , and q=2' and 5'
(k =r) where 10n =0, showing that indeed here only the last r digits are significant. Furthermore, since
the present results are equally valid for d or a ;j negative , we recover the cases q=ll (k=O) with
d=-10, and q=7,11 and 13 (k=O) with d=-103; see [1], (9.5). Other interesting examples are : q=37
with d=103, q=101 with d=-102
, q=73 and 137 with d=-10\ and q=41 and 271 with d=105 (all
k=O). Useful in computer applications, with d=16 , may be: q=3 ,5 and 15 if k=O, and the rich col
lection q=2,3 ,4,5,6,8,10,12,15,16,20, ... if k=l.
References
1 G.H. Hardy and E .M. Wright , An Introduction to the Theory of Numbers, 5th edition (1979),
Clarendon Press, Oxford.
2 L.E . Dickson, History of the Theory of Numbers, Volume I, Divisibility and Primality , 1952, Chel
sea Publishing Company, New York.
3 S.W. Rienstra, Problem 718, Nieuw Archief Voor Wiskunde (4), Vol. 2, no .3 (1984), and (4), Vol.
4, no.l (1986).
-- 3 --