2015-02-19-171735.sagews

Met priemtest2 wordt een geheel getal x aan de priemtest van Fermat onderworpen aan de hand van de vaste getuigen 2, 3, 4, ... 12. Als het getal x priem lijkt te zijn dan is de output x, zo niet is de output 0.

def priemtest(x,y):
    z= mod(y^(x-1)-1,x)
    return z
def priemtest2(e):
    q=[priemtest(e,x) for x in range (2,12)]
    f=sum(q)
    if f==0:
        return e
    else:
        return 0
priemtest2(97)

97

Berekening van de rij priemgetallen tussen de gehele getallen b en c.

def priemrij(b,c):
    r=[priemtest2(x) for x in range (b,c) if priemtest2(x) != 0]
    return r
priemrij(100,200)

[101, 103, 107, 109, 113, 127, 131, 135, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199]

Elk geheel getal g dat relatief priem is met x kan als getuige opgeroepen worden.

De getuigenrij van x beperkt zich tot alle getallen kleiner dan x en relatief priem met x.

def getuigen(x,g):
    if gcd(x,g)==1:
         return g
    else:
        return 0
getuigen(35,21)

0

def getuigenrij(x):
    r=[getuigen(x,g) for g in range (1,x) if getuigen(x,g) != 0]
    return r
getuigenrij(35)
getuigenrij (24)

[1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34] [1, 5, 7, 11, 13, 17, 19, 23]

Priemtest3 voert een Fermat-priemtest uit op een getal aan de hand van de hierboven gedefinieerde getuigenrij.

Indien het opgegeven getal x een priemgetal lijkt te zijn, is de output 0. In het andere geval is de output x.

def priemtest3(x):
    r=[priemtest(x,a) for a in getuigenrij(x)]
    f=max(r)
    return f

priemtest3 (529)
priemtest3 (143)
priemtest3 (97)

506 132 0

Een carmichaelgetal is een getal dat niet priem is en waarvoor alle getuigen uit de getuigenrij vals beweren dat dit getal wel priem is.

Met de instructie carmichaelrij kunnen alle carmichaelgetallen tussen bepaalde grenzen uitgefilterd worden.

def carmichael(x):
    if not is_prime(x) and priemtest3(x)==0:
        return x
    else:
        return 0
carmichael(561)
carmichael (529)

561 0

def carmichaelrij(b,c):
    r=[carmichael(x) for x in range (b,c) if carmichael(x)!=0]
    return r
carmichaelrij(550,600)

[561]

carmichaelrij(2,3000)

[561, 1105, 1729, 2465, 2821]

carmichaelrij(3000,6000)

[]

carmichaelrij(6000,10000)

[6601, 8911]

Het aantal getuigen dat opgeroepen kan worden voor de priemtest van Fermat voor x is het aantal gehele getallen kleiner dan x en relatief priem met x m.a.w. de totientwaarde van x of de euler_phi-waarde van x.

def aantalgetuigen(x):
    f=len(getuigenrij(x))
    return f
aantalgetuigen(35)
aantalgetuigen(703)

def aantalgetuigen2(x):
    r=euler_phi(x)
    return r
aantalgetuigen2(35)

24 648 24

def aantalvalsegetuigen(x):
    r=[1 for a in getuigenrij(x) if priemtest(x,a)==0]
    return sum(r)
aantalvalsegetuigen(703)

324

De halfcarmichaelgetallen zijn samengestelde gehele getallen waarvoor de helft van het aantal opgeroepen getuigen een valse priemgetuigenis aflegt voor de priemtest van Fermat. Dit betekent dat er een kans op twee is dat een geheel getal, relatief priem met x, ten onrechte beweert dat x priem is wanneer de congruentie van Fermat wordt uitgevoerd.

def halfcarm(x):
    if 2*(aantalvalsegetuigen(x))==(aantalgetuigen(x)) and not is_prime(x):
        return x
    else:
        return 0
halfcarm(703)
halfcarm(1891)

703 1891

def halfcarmrij(b,c):
    r=[halfcarm(x) for x in range (b,c) if halfcarm(x)!=0]
    return r
halfcarmrij(2,1000)
halfcarmrij(1000,1500)

[4, 6, 15, 91, 703] []

halfcarmrij(1500,3000)

[1891, 2701]

halfcarmrij(3000,6000)

[]

halfcarmrij(6000,7000)

[]

halfcarmrij(7000,8000)

[]

[n+10 for n in range (10000+10) if sum(q) == [mod((x+3)^(n+9)-1,n+10) for x in range (10)]]

[13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 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[n+10 for n in range (5000-10) if [0,0,0,0,0,0,0,0,0,0] == [mod((x+3)^(n+9)-1,n+10) for x in range (10)]]

[13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999]

def priemrij(b,c):
    r=[priemtest2(x) for x in range (b,c) if priemtest2(x)!=0]
    return r
priemrij(100,200)

Error in lines 4-4 Traceback (most recent call last): File "/projects/9f83a228-079b-408f-9ed0-f4d6c1937f46/.sagemathcloud/sage_server.py", line 875, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "", line 2, in priemrij NameError: global name 'priemtest2' is not defined