53 lines
1.9 KiB
Text
53 lines
1.9 KiB
Text
[[category:Prime Numbers]]
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In 1939, Jack Chernick proved that, for '''n ≥ 3''' and '''m ≥ 1''':
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U(n, m) = (6m + 1) * (12m + 1) * Product_{i=1..n-2} (2^i * 9m + 1)
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is a [https://en.wikipedia.org/wiki/Carmichael_number Carmichael number] if all the factors are primes and, for '''n > 4''', '''m''' is a multiple of '''2^(n-4)'''.
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;Example
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U(3, m) = (6m + 1) * (12m + 1) * (18m + 1)
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U(4, m) = U(3, m) * (2^2 * 9m + 1)
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U(5, m) = U(4, m) * (2^3 * 9m + 1)
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...
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U(n, m) = U(n-1, m) * (2^(n-2) * 9m + 1)
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* The smallest Chernick's Carmichael number with '''3''' prime factors, is: U(3, 1) = 1729.
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* The smallest Chernick's Carmichael number with '''4''' prime factors, is: U(4, 1) = 63973.
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* The smallest Chernick's Carmichael number with '''5''' prime factors, is: U(5, 380) = 26641259752490421121.
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For '''n = 5''', the smallest number '''m''' that satisfy Chernick's conditions, is '''m = 380''', therefore '''U(5, 380)''' is the smallest Chernick's Carmichael number with '''5''' prime factors.
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'''U(5, 380)''' is a Chernick's Carmichael number because '''m = 380''' is a multiple of '''2^(n-4)''', where '''n = 5''', and the factors { (6*380 + 1), (12*380 + 1), (18*380 + 1), (36*380 + 1), (72*380 + 1) } are all prime numbers.
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;Task
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For '''n ≥ 3''', let '''a(n)''' be the smallest Chernick's Carmichael number with '''n''' prime factors.
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* Compute '''a(n)''' for '''n = 3..9'''.
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* Optional: find '''a(10)'''.
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'''Note''': it's perfectly acceptable to show the terms in factorized form:
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a(3) = 7 * 13 * 19
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a(4) = 7 * 13 * 19 * 37
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a(5) = 2281 * 4561 * 6841 * 13681 * 27361
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...
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;See also
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* [http://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf Jack Chernick, On Fermat's simple theorem (PDF)]
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* [https://oeis.org/A318646 OEIS A318646: The least Chernick's "universal form" Carmichael number with n prime factors]
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; Related tasks
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* [[Carmichael 3 strong pseudoprimes]]
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<br><br>
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