翼年代记阿修罗王出场:孪生素数被证实了吗？

没有 否则怎么是世界三大数学难题之一？

据我所知，没有。

还没有，目前还是猜想，不过据说有重要突破，详细可以看一下网页：
http://www.chinainfo.gov.cn/data/200505/1_20050531_111338.html

没有

以36N（N+1）为界，孪生素数以波浪形式随着N的增大，渐渐增多。
1 7对
{5,7},{11,13},{17,19},{29,31},{41,43},{59,61},{71,73}
2 7对
{101,103},{107,109},{137,139},{149,151},{179,181},{191,193},{197,199},
3 8对
{227,229},{239,241},{269,271},{281,283},{311,313},{347,349},{419,421},{431,433},
4 7对
{461,463},{521,523},{569,571},{599,601},{617,619},{641,643},{659,661},
5 9对
{809,811},{821,823},{827,829},{857,859},{881,883},{1019,1021},{1031,1033},{1049,1051},{1061,1063},
6 11对
{1091,1093},{1151,1153},{1229,1231},{1277,1279},{1289,1291},{1301,1303},{1319,1321},{1427,1429},{1451,1453}, {1481,1483},{1487,1489},
7 11对
{1607,1609},{1619,1621},{1667,1669},{1697,1699},{1721,1723},{1787,1789},{1871,1873},{1877,1879},{1931,1933}, {1949,1951},{1997,1999},
8 13 对 {2027,2029},{2081,2083},{2087,2089},{2111,2113},{2129,2131},{2141,2143},{2237,2239},{2267,2269},{2309,2311},{2339,2341},{2381,2383},{2549,2551},{2591,2593},
9 10对
{2657,2659},{2687,2689},{2711,2713},{2729,2731},{2789,2791},{2801,2803},{2969,2971},{2999,3001},{3119,3121}, {3167,3169},
10 19 对 {3251,3253},{3257,3259},{3299,3301},{3329,3331},{3359,3361},{3371,3373},{3389,3391}{3461,3463},{3467,3469},{3527,3529},{3539,3541},{3557,3559},{3581,3583},{3671,3673},{3767,3769},{3821,3823},{3851,3853},{3917,3919},{3929,3931},
11 19对
{4001,4003},{4019,4021},{4049,4051},{4091,4093},{4127,4129},{4157,4159},{4217,4219},{4229,4231},{4241,4243}, {4259,4261},{4271,4273},{4337,4339},{4421,4423},{4481,4483},{4517,4519},{4547,4549},{4637,4639},{4649,4651}, {4721,4723},