Transcript Document

Discovery in Mathematics
an example
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1
Repeating decimal for 1/7
(click screen for the next step)
0.____________
142857142857…
7)1.000000000000…
7
30
28
20
14
60
56
40
35
50
49
1
2
1 = 0.142857142857142857 . . .
7
is a repeating decimal with a period of 6.
Can this statement be expressed mathematically?
1 = 142857
7 999999
(Proof involves the identity
where x = 10–6)
1_ = 1 + x + x2 + x3 + · ·
1–x ·
3
Is there an interesting question here?
one that generalizes from 1 = 142857
7 999999
(
)
999999 142857 (an integer)
=
7
107–1 – 1
= 142857 (an integer)
7
“For which ns does
10n–1 – 1
= integer ?”
n
4
(10n–1–1) / n
9 / 2 = 4.5
99 / 3 = 33
999 / 4 = 249.75
9999 / 5 = 1999.8
99999 / 6 = 16666.5
999999 / 7 = 142857
9999999 / 8 = 1249999.875
99999999 / 9 = 11111111
999999999 / 10 = 99999999.9
9999999999 / 11 = 909090909
99999999999 / 12 = 8333333333.25
999999999999 / 13 = 076923076823
9999999999999 / 14 = 714285714285.643
99999999999999 / 15 = 6666666666666.6
999999999999999 / 16 = 62499999999999.9
9999999999999999 / 17 = 588235294117647
99999999999999999 / 18 = 5555555555555555.5
999999999999999999 / 19 = 52631578947368421
9999999999999999999 / 20 = 499999999999999999.95
99999999999999999999 / 21 = 4761904761904761904.714
999999999999999999999 / 22 = 45454545454545454545.409
9999999999999999999999 / 23 = 434782608695652173913
5
Observations
10n–1 – 1
= integer
n
holds for all prime n except 2 and 5 (factors of 10).
10n–1 – 1
= integer
n
holds for all non-prime n, except 9 = 10 – 1.
Can we generalize further?
Are we just lucky that we use the base-10 system?
What about
an–1 – 1
= integer ?
n
6
a=2
(2n–1–1) / n
1 / 2 = 0.5
3/3=1
7 / 4 = 1.75
15 / 5 = 3
31 / 6 = 5.167
63 / 7 = 9
127 / 8 = 15.875
255 / 9 = 28.333
511 / 10 = 51.1
1023 / 11 = 93
2047 / 12 = 170.583
4095 / 13 = 315
8191 / 14 = 585.071
16383 / 15 = 1092.2
32767 / 16 = 2047.938
65535 / 17 = 3855
131071 / 18 = 7281.722
262143 / 19 = 13797
524287 / 20 = 26214.35
1048575 / 21 = 49932.143
2097151 / 22 = 95325.045
4194303 / 23 = 182361
7
Observations
2n–1 – 1
= integer
n
holds for all prime n except 2 (which is a
“factor” of a = 2).
2n–1 – 1
= integer
n
holds for non-prime n, from 2 to 23 at least.
But for n = 341 (a non-prime) we find that
2n–1 – 1
= integer
n
8
Conjecture (Guess)
n–1 – 1
2
based on
= integer observations?
n
an–1 – 1 integer
=
n
if n is prime and not a factor of a.
mod(an–1 – 1, n) = 0
if n is prime and not a factor of a.
This is “Fermat’s Little Theorem”
9
Usefulness of Fermat’s Little Theorem
Test for Primality
mod(an–1 – 1, n) = 0
almost only if n is prime and not a factor of a.
Allows “Public Key Encryption”
Pick p = 4099, q = 4111, m = 2 (p and q prime)
c = (p – 1)(q – 1)·m + 1 = 33685561
c = A·B, A = 2821 (public), B = 11941 (secret)
N = p·q = 16850989 (public)
x is the secret message
Encrypt: y = mod(xA, N),
Decrypt: x = mod(yB, N)
10