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A230579
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a(n) = 2^n mod 341.
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1
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1, 2, 4, 8, 16, 32, 64, 128, 256, 171, 1, 2, 4, 8, 16, 32, 64, 128, 256, 171, 1, 2, 4, 8, 16, 32, 64, 128, 256, 171, 1, 2, 4, 8, 16, 32, 64, 128, 256, 171, 1, 2, 4, 8, 16, 32, 64, 128, 256, 171, 1, 2, 4, 8, 16, 32, 64, 128, 256, 171, 1, 2, 4, 8, 16, 32, 64, 128, 256, 171
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OFFSET
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0,2
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COMMENTS
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Jeans asserts that it would have been impossible for the ancient Chinese to have discovered a case of failure for the converse of Fermat's little theorem because the smallest counterexample "(n = 341) consists of 103 figures" in base 10.
Granted that without a computer, the task of calculating 2^340 - 1 and dividing by 341 is tedious and error-prone, thus discouraging the discovery of that number as a counterexample to the so-called Chinese hypothesis.
But by instead computing just a few dozen powers of 2 modulo 341, it becomes readily apparent that the sequence of powers of 2 modulo 341 has a period of length 10 and therefore 2^340 = 1 mod 341, yet 341 = 11 * 31, which is not a prime number.
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LINKS
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FORMULA
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a(0) = 1, a(n) = 2*a(n-1) mod 341.
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EXAMPLE
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a(8) = 256 because 2^8 = 256.
a(9) = 171 because 2^9 = 512 and 512 - 341 = 171.
a(10) = 1 because 2 * 171 = 342 and 342 - 341 = 1.
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MATHEMATICA
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PowerMod[2, Range[0, 79], 341]
LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1}, {1, 2, 4, 8, 16, 32, 64, 128, 256}, 70] (* Ray Chandler, Jul 12 2015 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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