Learning Evidence 3 PI- Cipher PDF

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PI-CIPHER

First 1000 decimal places 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 The equation for pi decimals There are essentially 3 different methods to calculate pi to many decimals. One of the oldest is to use the power series expansion of atan(x) = x - x^3/3 + x^5/5 - ... together with formulas like pi = 16*atan(1/5) - 4*atan(1/239). This gives about 1.4 decimals per term. A second is to use formulas coming from Arithmetic-Geometric mean computations. A beautiful compendium of such formulas is given in the book pi and the AGM, (see references). They have the advantage of converging quadratically, i.e. you double the number of decimals per iteration. For instance, to obtain 1 000 000 decimals, around 20 iterations are sufficient. The disadvantage is that you need FFT type multiplication to get a reasonable speed, and this is not so easy to program. A third one comes from the theory of complex multiplication of elliptic curves, and was discovered by S. Ramanujan. This gives a number of beautiful formulas, but the most useful was missed by Ramanujan and discovered by the Chudnovsky's. It is the following (slightly modified for ease of programming): Set k_1 = 545140134; k_2 = 13591409; k_3 = 640320; k_4 = 100100025; k_5 = 327843840; k_6 = 53360; Then pi = (k_6 sqrt(k_3))/(S), where S = sum_(n = 0)^oo (-1)^n ((6n)!(k_2 + nk_1))/(n!^3(3n)!(8k_4k_5)^n) The great advantages of this formula are that 1) It converges linearly, but very fast (more than 14 decimal digits per term).

2) The way it is written, all operations to compute S can be programmed very simply. This is why the constant 8k_4k_5 appearing in the denominator has been written this way instead of 262537412640768000. This is how the Chudnovsky's have computed several billion decimals.

1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692

Q A Z W S X E D C R F V T G B Y H N U J M I K O L P...

To encrypt The numbers shown are the first 260 decimal places of pi grouped in 10s. The sequence of letters shown can be repeated after it hits every 26th group of pi decimal numbers if the sender wants to make the cipher a bit more complex, but in order to do that, the sender should calculate for the decimal group of numbers of pi in order to get that desired outcome, but this can be optional. The arrangement of letters is based from the keyboard by pressing each key from left to right and in a top to bottom order. The sender must calculate the desired group of decimals in order to get the corresponding letter for the group. The sender may also include the preferred amount of decimal places used to encrypt the message to lessen the complexity of the decryption. pi260 - (the preferred amount of decimal places used to encrypt the message) 4428810975 8979323846 0938446095 8410270193

6659334461 6939937510 8214808651 6446229489 8521105559

To decrypt The receiver would also have to calculate the given amount of decimal places set by the sender then group them into 10s and assign the alphabet of the keyboard through a top to bottom and in a left to right manner. Now that the arrangement of letter has been placed, the receiver can now identify the letters encrypted by the sender. If the sender did not provide a preferred amount of decimal places, then the sender might have intended to make the decryption a bit more complex for the receiver. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692

Q A Z W S X E D C R F V T G B Y H N U J M I K O L P

Using this given decimal places of pi, we can now determine what was the encrypted message. pi260 - (the preferred amount of decimal places used to encrypt the message) 4428810975 8979323846 0938446095 8410270193 6659334461 6939937510 8214808651 6446229489 8521105559 =

MATH IS FUN

Next example: Can you guess it? 8628034825 8410270193 3421170679 6659334461 6939937510 0938446095 4428810975 8979323846 6939937510 0938446095 6659334461 4428810975 5923078164!

= CHRISTMAS TIME!

For the last example, write down in the paper and try to guess what is the message mean. 8628034825 5923078164 2712019091 5923078164 5359408128 3421170679 8979323846 0938446095 5923078164 0938446095 8410270193 5923078164 2712019091 3786783165 3282306647 5923078164 3786783165 8214808651 5058223172 3786783165 0628620899

= CELEBRATE THE LOVE OF GOD

Link: https://www.youtube.com/watch?v=kvJyKrqXaSc&feature=youtu.be...