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},{