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Discovery in Mathematicsan example

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2

7)1.000000000000…0.____________1

73

4

028 2

2

014 6

8

056 4

5

035 5

7

049 1

142857…

Repeating decimal for 1/7(click screen for the next step)

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17

= 0.142857142857142857 . . .

is a repeating decimal with a period of 6. Can this statement be expressed mathematically?

17

= 142857999999

(Proof involves the identity

where x = 10–6)

1_1 – x

= 1 + x + x2 + x3 + · · ·

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Is there an interesting question here?

10n–1 – 1 n

= integer ?”

“For which ns does

999999

7

= 142857 (an integer)

107–1 – 1 7

= 142857 (an integer)

one that generalizes from 17

= 142857999999( )

5

(10n–1–1) / n

9 / 2 = 4.599 / 3 = 33999 / 4 = 249.759999 / 5 = 1999.899999 / 6 = 16666.5999999 / 7 = 1428579999999 / 8 = 1249999.87599999999 / 9 = 11111111999999999 / 10 = 99999999.99999999999 / 11 = 90909090999999999999 / 12 = 8333333333.25999999999999 / 13 = 0769230768239999999999999 / 14 = 714285714285.64399999999999999 / 15 = 6666666666666.6999999999999999 / 16 = 62499999999999.99999999999999999 / 17 = 58823529411764799999999999999999 / 18 = 5555555555555555.5999999999999999999 / 19 = 526315789473684219999999999999999999 / 20 = 499999999999999999.9599999999999999999999 / 21 = 4761904761904761904.714999999999999999999999 / 22 = 45454545454545454545.4099999999999999999999999 / 23 = 434782608695652173913

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10n–1 – 1 n

= integer

holds for all prime n except 2 and 5 (factors of 10).

= 10 – 1.

Can we generalize further?

Are we just lucky that we use the base-10 system? What about

an–1 – 1 n

= integer ?

Observations

holds for all non-prime n, except 9

10n–1 – 1 n

= integer

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(2n–1–1) / n

1 / 2 = 0.53 / 3 = 17 / 4 = 1.7515 / 5 = 331 / 6 = 5.16763 / 7 = 9127 / 8 = 15.875255 / 9 = 28.333511 / 10 = 51.11023 / 11 = 932047 / 12 = 170.5834095 / 13 = 3158191 / 14 = 585.07116383 / 15 = 1092.232767 / 16 = 2047.93865535 / 17 = 3855131071 / 18 = 7281.722262143 / 19 = 13797524287 / 20 = 26214.351048575 / 21 = 49932.1432097151 / 22 = 95325.0454194303 / 23 = 182361

a = 2

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2n–1 – 1 n

= integer

But for n = 341 (a non-prime) we find that

holds for all prime n except 2 (which is a “factor” of a = 2).

2n–1 – 1 n

= integer

Observations

holds for non-prime n, from 2 to 23 at least.

2n–1 – 1 n

= integer

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Conjecture (Guess)

mod(an–1 – 1, n) = 0if n is prime and not a factor of a.

This is “Fermat’s Little Theorem”

2n–1 – 1 n

= integer observations?based on

an–1 – 1 n

if n is prime and not a factor of a.

= integer

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Usefulness of Fermat’s Little Theorem

Test for Primality

mod(an–1 – 1, n) = 0almost 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 messageEncrypt: y = mod(xA, N), Decrypt: x = mod(yB, N)