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1 Discovery in Mathematics an example (Click anywhere on the page)

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2 7) … 0.____________ … Repeating decimal for 1/7 (click screen for the next step)

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= is a repeating decimal with a period of 6. Can this statement be expressed mathematically? 1717 = (Proof involves the identity where x = 10 –6 ) 1_ 1 – x = 1 + x + x 2 + x 3 + · · ·

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4 Is there an interesting question here? 10 n–1 – 1 n = integer ?” “For which ns does = (an integer) 10 7–1 – 1 7 = (an integer) one that generalizes from 1717 = ()

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5 (10 n–1 –1) / n 9 / 2 = / 3 = / 4 = / 5 = / 6 = / 7 = / 8 = / 9 = / 10 = / 11 = / 12 = / 13 = / 14 = / 15 = / 16 = / 17 = / 18 = / 19 = / 20 = / 21 = / 22 = / 23 =

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6 10 n–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 a n–1 – 1 n = integer ? Observations holds for all non-prime n, except 9 10 n–1 – 1 n = integer

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7 (2 n–1 –1) / n 1 / 2 = / 3 = 1 7 / 4 = / 5 = 3 31 / 6 = / 7 = / 8 = / 9 = / 10 = / 11 = / 12 = / 13 = / 14 = / 15 = / 16 = / 17 = / 18 = / 19 = / 20 = / 21 = / 22 = / 23 = a = 2

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8 2 n–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). 2 n–1 – 1 n = integer Observations holds for non-prime n, from 2 to 23 at least. 2 n–1 – 1 n = integer

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9 Conjecture (Guess) mod(a n–1 – 1, n) = 0 if n is prime and not a factor of a. This is “Fermat’s Little Theorem” 2 n–1 – 1 n = integer observations?based on a n–1 – 1 n if n is prime and not a factor of a. = integer

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10 Usefulness of Fermat’s Little Theorem Test for Primality mod(a n–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 = c = A·B, A = 2821 (public), B = (secret) N = p·q = (public) x is the secret message Encrypt: y = mod(x A, N), Decrypt: x = mod(y B, N)

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