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

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2 7)1.000000000000… 0.____________173173 4 0 28 2 2 0 14 6 8 0 56 4 5 0 35 5 7 0 49 1 142857… Repeating decimal for 1/7 (click screen for the next step)

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3 1717 = 0.142857142857142857... is a repeating decimal with a period of 6. Can this statement be expressed mathematically? 1717 = 142857 999999 (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 999999 7 = 142857 (an integer) 10 7–1 – 1 7 = 142857 (an integer) one that generalizes from 1717 = 142857 999999 ()

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5 (10 n–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

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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 = 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 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 = 33685561 c = A·B, A = 2821 (public), B = 11941 (secret) N = p·q = 16850989 (public) x is the secret message Encrypt: y = mod(x A, N), Decrypt: x = mod(y B, N)

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Mathematical Induction. F(1) = 1; F(n+1) = F(n) + (2n+1) for n≥1 100816449362516941 10987654321 F(n) n F(n) =n 2 for all n ≥ 1 Prove it!

Mathematical Induction. F(1) = 1; F(n+1) = F(n) + (2n+1) for n≥1 100816449362516941 10987654321 F(n) n F(n) =n 2 for all n ≥ 1 Prove it!

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