p1,p2是素数,p(n+2)是p(n+1)+p(n)+2000的最大素因子，试证{p(n)}有界

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推论，这个序列只含有限个素数，并且从某项开始具有周期性

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Python 程序：

1. from math import *
   
2. 
   
3. def maxPrimeFactor(n):
   
4.    P = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,\
   
5.    73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,\
   
6.    163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,\
   
7.    251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,\
   
8.    349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,\
   
9.    443,449,457,461,463,467,479,487,491,499,503,509,521,523,541]
   
10.    p = 1
    
11.    i = 0
    
12.    d = P[0]
    
13.    s = int(sqrt(n))
    
14.    while n > 1:
    
15.        if n % d == 0:
    
16.            p = d
    
17.            while n % d == 0:
    
18.                n /= d
    
19.            s = int(sqrt(n))
    
20.        i = i + 1
    
21. &#160; &#160; &#160; &#160;if i < 100: d = P[i]
    
22. &#160; &#160; &#160; &#160;else: d = d + 2
    
23. &#160; &#160; &#160; &#160;if d > s:
    
24. &#160; &#160; &#160; &#160; &#160; &#160;if n > 1: p = n
    
25. &#160; &#160; &#160; &#160; &#160; &#160;break
    
26. &#160; &#160;return p
    
27. 
    
28. def pPlusList(a,b,c=2000):
    
29. &#160; &#160;L = [a,b]
    
30. &#160; &#160;while True:
    
31. &#160; &#160; &#160; &#160;d = maxPrimeFactor(a + b + c)
    
32. &#160; &#160; &#160; &#160;e = maxPrimeFactor(b + c + d)
    
33. &#160; &#160; &#160; &#160;m = len(L)
    
34. &#160; &#160; &#160; &#160;i = 0
    
35. &#160; &#160; &#160; &#160;while i < m:
    
36. &#160; &#160; &#160; &#160; &#160; &#160;if L[i] == d:
    
37. &#160; &#160; &#160; &#160; &#160; &#160;if i+1 < m and L[i+1] == e:
    
38. &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160;L.append(d)
    
39. &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160;L.append(e)
    
40. &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160;return L,i
    
41. &#160; &#160; &#160; &#160; &#160; &#160;i = i + 1
    
42. &#160; &#160; &#160; &#160;L.append(d)
    
43. &#160; &#160; &#160; &#160;L.append(e)
    
44. &#160; &#160; &#160; &#160;a = d
    
45. &#160; &#160; &#160; &#160;b = e
    

复制代码

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以任意自然数取代 2000， 首项及次项不必为素数，主贴的结论仍成立。