《数论探秘》中对杰波夫猜想的证明

ysr

   杰波夫猜想是数论中的一个未解难题，也被称为勒让德猜想 。该猜想认为，对于任一正整数n，在n^2和(n+1)^2之间至少存在两个素数；而与之相关的勒让德猜想仅断言至少存在一个素数，因此杰波夫猜想是更强的命题。该猜想由杰波夫于1855年提出。
   《数论探秘》中的描述：数列n^2+n;n^2+n-1;……n^2这n+1项中,至少有一项乘4加1为质数,至少有一项乘4加3为质数;如6,5,4之中4*4+1=17为质数。
     证明：
      引理：同余性质定理（孙子定理）：整数除法中余数具有以下基本性质：
（1）被除数增加或减去除数的倍数，余数不变；
（2）被除数扩大或缩小几倍，除数不变，则余数扩大或缩小相同倍数。
       设数列n^2+n,n^2+n-1,n^2+n-2,……，n^2,共n+1项，即下面表1中各纵列平方数上方的部分，设前n项*4+1能被2m+1之内(>=3)的n个奇数整除。
3<=2m+1<=2n+1,设这前n项除以2m+1的余数为R（代表n个不同值），R不等于0且4R+1能被2m+1整除，那么：欲使第n+1项*4+1即4n^2+1为合数，则须：n^2+n=n^2=R MOD(2m+1),当且仅当R=0,即 n=0 MOD(2m+1)时该式成立,换句话说，仅当n能被2m+1整除时成立，而 R不等于0，故由同余式性质定理，即孙子定理的性质定理得该式不成立。
则此时4n^2+1必为素数。
欲使4n^2+1为合数,则须前n项中至少有一项*4+1为素数，则证明该数列中至少一项*4+1为素数，*4+3也是至少一项为素数，原理同上，证毕！
此俩素数是落在杰波夫猜想的区间内的，所以杰波夫猜想成立。
勒让德猜想：n为大于等于1的正整数，在区间（n^2,(n+1)^2）内至少有一个素数。
   举例：
   n=2时，有：2*2+2=6
2*2+2-1=5
2*2+2-2=4
则有：4*6+1=25=5*5，4*6+3=27=3*3*3
4*5+1=21=3*7，4*5+3=23是素数
而4*4+1=17是素数，4*4+3=19是素数

同理，从小到大排列也成立，如：n^2+1,n^2+2,……，n^2+n,n^2+n+1,这n+1项中若前n项*4+1全是合数，则第n+1项*4+1必是素数。
  举例：
n=2时，有：2*2+1=5，2*2+2=6，2*2+3=7
4*5+1=21=3*7，4*6+1=25=5*5，而4*7+1=29是素数。
4*5+3=23是素数，4*6+3=27=3*3*3，4*7+3=31是素数。

ysr

设n≥2，数列n^2+1,n^2+2,……,n^2+n这n项*4+1全是合数的情况，上面已经有一个例子就是n=2的时候，下面给出另一个例子：
当n=7的时候：
7*7+1=50，4*50+1=201=3*67
7*7+2=51，4*51+1=205=5*41
7*7+3=52，4*52+1=209=11*19
7*7+4=53，4*53+1=213=3*71
7*7+5=54，4*54+1=217=7*31
7*7+6=55，4*55+1=221=13*17
7*7+7=56，4*56+1=225=3*3*5*5
而7*7+8=57，4*57+1=229是素数
7*7=49，4*49+1=197是素数

ysr

Private Sub Command1_Click()
Dim k, n
n = Val(Text1)
k = 1
Do While k <= n
m = Val(n ^ 2 + k)
p = Val(4 * m + 1)
f = fenjieyinzi(Val(p))
s1 = s1 & n & " ^2+ " & k & " = " & m & ", 4* " & m & " +1 =" & p & " = " & f & vbCrLf
k = k + 1
Loop
Text2 = s1

End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""

End Sub

Private Function fenjieyinzi(sa As String) As String
Dim x, a, b, k As String
a = Val(sa)

x = 3
If a <= 1 Or a > Int(a) Then
If a = 1 Then
fenjieyinzi = "它既不是质数，也不是合数"

Else
MsgBox "error"
End If
  
Else

Do While a / 2 = Int(a / 2) And a >= 4
  
If b = 0 Then
fenjieyinzi = fenjieyinzi & "2"
b = 1
Else
fenjieyinzi = fenjieyinzi & "*2"
End If
a = a / 2
k = a
  
Loop

Do While a > 1
Do While x <= Sqr(a)
Do While a / x = Int(a / x) And a >= x * x
  
If b = 0 Then
fenjieyinzi = fenjieyinzi & x
b = 1
Else
fenjieyinzi = fenjieyinzi & "*" & x
End If
a = a / x
Loop
  
x = x + 2
Loop
  
k = a
a = 1
Loop
  
If b = 1 Then
fenjieyinzi = fenjieyinzi & "*" & k
Else
fenjieyinzi = "这是一个质数"
End If
  
End If

End Function

ysr

80^2+1 = 6401, 4*6401 +1 = 25605 = 3*3*5*569
80^2+2 = 6402, 4*6402 +1 = 25609 = 这是一个质数
80^2+3 = 6403, 4*6403 +1 = 25613 = 7*3659
80^2+4 = 6404, 4*6404 +1 = 25617 = 3*8539
80^2+5 = 6405, 4*6405 +1 = 25621 = 这是一个质数
80^2+6 = 6406, 4*6406 +1 = 25625 = 5*5*5*5*41
80^2+7 = 6407, 4*6407 +1 = 25629 = 3*8543
80^2+8 = 6408, 4*6408 +1 = 25633 = 这是一个质数
80^2+9 = 6409, 4*6409 +1 = 25637 = 31*827
80^2+10 = 6410, 4*6410 +1 = 25641 = 3*3*7*11*37
80^2+11 = 6411, 4*6411 +1 = 25645 = 5*23*223
80^2+12 = 6412, 4*6412 +1 = 25649 = 13*1973
80^2+13 = 6413, 4*6413 +1 = 25653 = 3*17*503
80^2+14 = 6414, 4*6414 +1 = 25657 = 这是一个质数
80^2+15 = 6415, 4*6415 +1 = 25661 = 67*383
80^2+16 = 6416, 4*6416 +1 = 25665 = 3*5*29*59
80^2+17 = 6417, 4*6417 +1 = 25669 = 7*19*193
80^2+18 = 6418, 4*6418 +1 = 25673 = 这是一个质数
80^2+19 = 6419, 4*6419 +1 = 25677 = 3*3*3*3*317
80^2+20 = 6420, 4*6420 +1 = 25681 = 61*421
80^2+21 = 6421, 4*6421 +1 = 25685 = 5*11*467
80^2+22 = 6422, 4*6422 +1 = 25689 = 3*8563
80^2+23 = 6423, 4*6423 +1 = 25693 = 这是一个质数
80^2+24 = 6424, 4*6424 +1 = 25697 = 7*3671
80^2+25 = 6425, 4*6425 +1 = 25701 = 3*13*659
80^2+26 = 6426, 4*6426 +1 = 25705 = 5*53*97
80^2+27 = 6427, 4*6427 +1 = 25709 = 47*547
80^2+28 = 6428, 4*6428 +1 = 25713 = 3*3*2857
80^2+29 = 6429, 4*6429 +1 = 25717 = 这是一个质数
80^2+30 = 6430, 4*6430 +1 = 25721 = 17*17*89
80^2+31 = 6431, 4*6431 +1 = 25725 = 3*5*5*7*7*7
80^2+32 = 6432, 4*6432 +1 = 25729 = 11*2339
80^2+33 = 6433, 4*6433 +1 = 25733 = 这是一个质数
80^2+34 = 6434, 4*6434 +1 = 25737 = 3*23*373
80^2+35 = 6435, 4*6435 +1 = 25741 = 这是一个质数
80^2+36 = 6436, 4*6436 +1 = 25745 = 5*19*271
80^2+37 = 6437, 4*6437 +1 = 25749 = 3*3*2861
80^2+38 = 6438, 4*6438 +1 = 25753 = 7*13*283
80^2+39 = 6439, 4*6439 +1 = 25757 = 43*599
80^2+40 = 6440, 4*6440 +1 = 25761 = 3*31*277
80^2+41 = 6441, 4*6441 +1 = 25765 = 5*5153
80^2+42 = 6442, 4*6442 +1 = 25769 = 73*353
80^2+43 = 6443, 4*6443 +1 = 25773 = 3*11*11*71
80^2+44 = 6444, 4*6444 +1 = 25777 = 149*173
80^2+45 = 6445, 4*6445 +1 = 25781 = 7*29*127
80^2+46 = 6446, 4*6446 +1 = 25785 = 3*3*3*5*191
80^2+47 = 6447, 4*6447 +1 = 25789 = 17*37*41
80^2+48 = 6448, 4*6448 +1 = 25793 = 这是一个质数
80^2+49 = 6449, 4*6449 +1 = 25797 = 3*8599
80^2+50 = 6450, 4*6450 +1 = 25801 = 这是一个质数
80^2+51 = 6451, 4*6451 +1 = 25805 = 5*13*397
80^2+52 = 6452, 4*6452 +1 = 25809 = 3*7*1229
80^2+53 = 6453, 4*6453 +1 = 25813 = 83*311
80^2+54 = 6454, 4*6454 +1 = 25817 = 11*2347
80^2+55 = 6455, 4*6455 +1 = 25821 = 3*3*19*151
80^2+56 = 6456, 4*6456 +1 = 25825 = 5*5*1033
80^2+57 = 6457, 4*6457 +1 = 25829 = 23*1123
80^2+58 = 6458, 4*6458 +1 = 25833 = 3*79*109
80^2+59 = 6459, 4*6459 +1 = 25837 = 7*3691
80^2+60 = 6460, 4*6460 +1 = 25841 = 这是一个质数
80^2+61 = 6461, 4*6461 +1 = 25845 = 3*5*1723
80^2+62 = 6462, 4*6462 +1 = 25849 = 这是一个质数
80^2+63 = 6463, 4*6463 +1 = 25853 = 103*251
80^2+64 = 6464, 4*6464 +1 = 25857 = 3*3*13*13*17
80^2+65 = 6465, 4*6465 +1 = 25861 = 11*2351
80^2+66 = 6466, 4*6466 +1 = 25865 = 5*7*739
80^2+67 = 6467, 4*6467 +1 = 25869 = 3*8623
80^2+68 = 6468, 4*6468 +1 = 25873 = 这是一个质数
80^2+69 = 6469, 4*6469 +1 = 25877 = 113*229
80^2+70 = 6470, 4*6470 +1 = 25881 = 3*8627
80^2+71 = 6471, 4*6471 +1 = 25885 = 5*31*167
80^2+72 = 6472, 4*6472 +1 = 25889 = 这是一个质数
80^2+73 = 6473, 4*6473 +1 = 25893 = 3*3*3*7*137
80^2+74 = 6474, 4*6474 +1 = 25897 = 19*29*47
80^2+75 = 6475, 4*6475 +1 = 25901 = 59*439
80^2+76 = 6476, 4*6476 +1 = 25905 = 3*5*11*157
80^2+77 = 6477, 4*6477 +1 = 25909 = 13*1993
80^2+78 = 6478, 4*6478 +1 = 25913 = 这是一个质数
80^2+79 = 6479, 4*6479 +1 = 25917 = 3*53*163
80^2+80 = 6480, 4*6480 +1 = 25921 = 7*7*23*23

ysr

81 ^2+ 1 = 6562, 4* 6562 +1 =26249 = 这是一个质数
81 ^2+ 2 = 6563, 4* 6563 +1 =26253 = 3*3*2917
81 ^2+ 3 = 6564, 4* 6564 +1 =26257 = 7*11*11*31
81 ^2+ 4 = 6565, 4* 6565 +1 =26261 = 这是一个质数
81 ^2+ 5 = 6566, 4* 6566 +1 =26265 = 3*5*17*103
81 ^2+ 6 = 6567, 4* 6567 +1 =26269 = 109*241
81 ^2+ 7 = 6568, 4* 6568 +1 =26273 = 13*43*47
81 ^2+ 8 = 6569, 4* 6569 +1 =26277 = 3*19*461
81 ^2+ 9 = 6570, 4* 6570 +1 =26281 = 41*641
81 ^2+ 10 = 6571, 4* 6571 +1 =26285 = 5*7*751
81 ^2+ 11 = 6572, 4* 6572 +1 =26289 = 3*3*23*127
81 ^2+ 12 = 6573, 4* 6573 +1 =26293 = 这是一个质数
81 ^2+ 13 = 6574, 4* 6574 +1 =26297 = 这是一个质数
81 ^2+ 14 = 6575, 4* 6575 +1 =26301 = 3*11*797
81 ^2+ 15 = 6576, 4* 6576 +1 =26305 = 5*5261
81 ^2+ 16 = 6577, 4* 6577 +1 =26309 = 这是一个质数
81 ^2+ 17 = 6578, 4* 6578 +1 =26313 = 3*7*7*179
81 ^2+ 18 = 6579, 4* 6579 +1 =26317 = 这是一个质数
81 ^2+ 19 = 6580, 4* 6580 +1 =26321 = 这是一个质数
81 ^2+ 20 = 6581, 4* 6581 +1 =26325 = 3*3*3*3*5*5*13
81 ^2+ 21 = 6582, 4* 6582 +1 =26329 = 113*233
81 ^2+ 22 = 6583, 4* 6583 +1 =26333 = 17*1549
81 ^2+ 23 = 6584, 4* 6584 +1 =26337 = 3*8779
81 ^2+ 24 = 6585, 4* 6585 +1 =26341 = 7*53*71
81 ^2+ 25 = 6586, 4* 6586 +1 =26345 = 5*11*479
81 ^2+ 26 = 6587, 4* 6587 +1 =26349 = 3*8783
81 ^2+ 27 = 6588, 4* 6588 +1 =26353 = 19*19*73
81 ^2+ 28 = 6589, 4* 6589 +1 =26357 = 这是一个质数
81 ^2+ 29 = 6590, 4* 6590 +1 =26361 = 3*3*29*101
81 ^2+ 30 = 6591, 4* 6591 +1 =26365 = 5*5273
81 ^2+ 31 = 6592, 4* 6592 +1 =26369 = 7*3767
81 ^2+ 32 = 6593, 4* 6593 +1 =26373 = 3*59*149
81 ^2+ 33 = 6594, 4* 6594 +1 =26377 = 13*2029
81 ^2+ 34 = 6595, 4* 6595 +1 =26381 = 23*31*37
81 ^2+ 35 = 6596, 4* 6596 +1 =26385 = 3*5*1759
81 ^2+ 36 = 6597, 4* 6597 +1 =26389 = 11*2399
81 ^2+ 37 = 6598, 4* 6598 +1 =26393 = 这是一个质数
81 ^2+ 38 = 6599, 4* 6599 +1 =26397 = 3*3*7*419
81 ^2+ 39 = 6600, 4* 6600 +1 =26401 = 17*1553
81 ^2+ 40 = 6601, 4* 6601 +1 =26405 = 5*5281
81 ^2+ 41 = 6602, 4* 6602 +1 =26409 = 3*8803
81 ^2+ 42 = 6603, 4* 6603 +1 =26413 = 61*433
81 ^2+ 43 = 6604, 4* 6604 +1 =26417 = 这是一个质数
81 ^2+ 44 = 6605, 4* 6605 +1 =26421 = 3*8807
81 ^2+ 45 = 6606, 4* 6606 +1 =26425 = 5*5*7*151
81 ^2+ 46 = 6607, 4* 6607 +1 =26429 = 13*19*107
81 ^2+ 47 = 6608, 4* 6608 +1 =26433 = 3*3*3*11*89
81 ^2+ 48 = 6609, 4* 6609 +1 =26437 = 这是一个质数
81 ^2+ 49 = 6610, 4* 6610 +1 =26441 = 137*193
81 ^2+ 50 = 6611, 4* 6611 +1 =26445 = 3*5*41*43
81 ^2+ 51 = 6612, 4* 6612 +1 =26449 = 这是一个质数
81 ^2+ 52 = 6613, 4* 6613 +1 =26453 = 7*3779
81 ^2+ 53 = 6614, 4* 6614 +1 =26457 = 3*8819
81 ^2+ 54 = 6615, 4* 6615 +1 =26461 = 47*563
81 ^2+ 55 = 6616, 4* 6616 +1 =26465 = 5*67*79
81 ^2+ 56 = 6617, 4* 6617 +1 =26469 = 3*3*17*173
81 ^2+ 57 = 6618, 4* 6618 +1 =26473 = 23*1151
81 ^2+ 58 = 6619, 4* 6619 +1 =26477 = 11*29*83
81 ^2+ 59 = 6620, 4* 6620 +1 =26481 = 3*7*13*97
81 ^2+ 60 = 6621, 4* 6621 +1 =26485 = 5*5297
81 ^2+ 61 = 6622, 4* 6622 +1 =26489 = 这是一个质数
81 ^2+ 62 = 6623, 4* 6623 +1 =26493 = 3*8831
81 ^2+ 63 = 6624, 4* 6624 +1 =26497 = 这是一个质数
81 ^2+ 64 = 6625, 4* 6625 +1 =26501 = 这是一个质数
81 ^2+ 65 = 6626, 4* 6626 +1 =26505 = 3*3*5*19*31
81 ^2+ 66 = 6627, 4* 6627 +1 =26509 = 7*7*541
81 ^2+ 67 = 6628, 4* 6628 +1 =26513 = 这是一个质数
81 ^2+ 68 = 6629, 4* 6629 +1 =26517 = 3*8839
81 ^2+ 69 = 6630, 4* 6630 +1 =26521 = 11*2411
81 ^2+ 70 = 6631, 4* 6631 +1 =26525 = 5*5*1061
81 ^2+ 71 = 6632, 4* 6632 +1 =26529 = 3*37*239
81 ^2+ 72 = 6633, 4* 6633 +1 =26533 = 13*13*157
81 ^2+ 73 = 6634, 4* 6634 +1 =26537 = 7*17*223
81 ^2+ 74 = 6635, 4* 6635 +1 =26541 = 3*3*3*983
81 ^2+ 75 = 6636, 4* 6636 +1 =26545 = 5*5309
81 ^2+ 76 = 6637, 4* 6637 +1 =26549 = 139*191
81 ^2+ 77 = 6638, 4* 6638 +1 =26553 = 3*53*167
81 ^2+ 78 = 6639, 4* 6639 +1 =26557 = 这是一个质数
81 ^2+ 79 = 6640, 4* 6640 +1 =26561 = 这是一个质数
81 ^2+ 80 = 6641, 4* 6641 +1 =26565 = 3*5*7*11*23
81 ^2+ 81 = 6642, 4* 6642 +1 =26569 = 163*163

ysr

100 ^2+ 1 = 10001, 4* 10001 +1 =40005 = 3*3*5*7*127
100 ^2+ 2 = 10002, 4* 10002 +1 =40009 = 这是一个质数
100 ^2+ 3 = 10003, 4* 10003 +1 =40013 = 这是一个质数
100 ^2+ 4 = 10004, 4* 10004 +1 =40017 = 3*13339
100 ^2+ 5 = 10005, 4* 10005 +1 =40021 = 31*1291
100 ^2+ 6 = 10006, 4* 10006 +1 =40025 = 5*5*1601
100 ^2+ 7 = 10007, 4* 10007 +1 =40029 = 3*11*1213
100 ^2+ 8 = 10008, 4* 10008 +1 =40033 = 7*7*19*43
100 ^2+ 9 = 10009, 4* 10009 +1 =40037 = 这是一个质数
100 ^2+ 10 = 10010, 4* 10010 +1 =40041 = 3*3*3*1483
100 ^2+ 11 = 10011, 4* 10011 +1 =40045 = 5*8009
100 ^2+ 12 = 10012, 4* 10012 +1 =40049 = 29*1381
100 ^2+ 13 = 10013, 4* 10013 +1 =40053 = 3*13*13*79
100 ^2+ 14 = 10014, 4* 10014 +1 =40057 = 41*977
100 ^2+ 15 = 10015, 4* 10015 +1 =40061 = 7*59*97
100 ^2+ 16 = 10016, 4* 10016 +1 =40065 = 3*5*2671
100 ^2+ 17 = 10017, 4* 10017 +1 =40069 = 17*2357
100 ^2+ 18 = 10018, 4* 10018 +1 =40073 = 11*3643
100 ^2+ 19 = 10019, 4* 10019 +1 =40077 = 3*3*61*73
100 ^2+ 20 = 10020, 4* 10020 +1 =40081 = 149*269
100 ^2+ 21 = 10021, 4* 10021 +1 =40085 = 5*8017
100 ^2+ 22 = 10022, 4* 10022 +1 =40089 = 3*7*23*83
100 ^2+ 23 = 10023, 4* 10023 +1 =40093 = 这是一个质数
100 ^2+ 24 = 10024, 4* 10024 +1 =40097 = 101*397
100 ^2+ 25 = 10025, 4* 10025 +1 =40101 = 3*13367
100 ^2+ 26 = 10026, 4* 10026 +1 =40105 = 5*13*617
100 ^2+ 27 = 10027, 4* 10027 +1 =40109 = 19*2111
100 ^2+ 28 = 10028, 4* 10028 +1 =40113 = 3*3*4457
100 ^2+ 29 = 10029, 4* 10029 +1 =40117 = 7*11*521
100 ^2+ 30 = 10030, 4* 10030 +1 =40121 = 53*757
100 ^2+ 31 = 10031, 4* 10031 +1 =40125 = 3*5*5*5*107
100 ^2+ 32 = 10032, 4* 10032 +1 =40129 = 这是一个质数
100 ^2+ 33 = 10033, 4* 10033 +1 =40133 = 67*599
100 ^2+ 34 = 10034, 4* 10034 +1 =40137 = 3*17*787
100 ^2+ 35 = 10035, 4* 10035 +1 =40141 = 137*293
100 ^2+ 36 = 10036, 4* 10036 +1 =40145 = 5*7*31*37
100 ^2+ 37 = 10037, 4* 10037 +1 =40149 = 3*3*3*1487
100 ^2+ 38 = 10038, 4* 10038 +1 =40153 = 这是一个质数
100 ^2+ 39 = 10039, 4* 10039 +1 =40157 = 13*3089
100 ^2+ 40 = 10040, 4* 10040 +1 =40161 = 3*11*1217
100 ^2+ 41 = 10041, 4* 10041 +1 =40165 = 5*29*277
100 ^2+ 42 = 10042, 4* 10042 +1 =40169 = 这是一个质数
100 ^2+ 43 = 10043, 4* 10043 +1 =40173 = 3*7*1913
100 ^2+ 44 = 10044, 4* 10044 +1 =40177 = 这是一个质数
100 ^2+ 45 = 10045, 4* 10045 +1 =40181 = 23*1747
100 ^2+ 46 = 10046, 4* 10046 +1 =40185 = 3*3*5*19*47
100 ^2+ 47 = 10047, 4* 10047 +1 =40189 = 这是一个质数
100 ^2+ 48 = 10048, 4* 10048 +1 =40193 = 这是一个质数
100 ^2+ 49 = 10049, 4* 10049 +1 =40197 = 3*13399
100 ^2+ 50 = 10050, 4* 10050 +1 =40201 = 7*5743
100 ^2+ 51 = 10051, 4* 10051 +1 =40205 = 5*11*17*43
100 ^2+ 52 = 10052, 4* 10052 +1 =40209 = 3*13*1031
100 ^2+ 53 = 10053, 4* 10053 +1 =40213 = 这是一个质数
100 ^2+ 54 = 10054, 4* 10054 +1 =40217 = 131*307
100 ^2+ 55 = 10055, 4* 10055 +1 =40221 = 3*3*41*109
100 ^2+ 56 = 10056, 4* 10056 +1 =40225 = 5*5*1609
100 ^2+ 57 = 10057, 4* 10057 +1 =40229 = 7*7*821
100 ^2+ 58 = 10058, 4* 10058 +1 =40233 = 3*13411
100 ^2+ 59 = 10059, 4* 10059 +1 =40237 = 这是一个质数
100 ^2+ 60 = 10060, 4* 10060 +1 =40241 = 这是一个质数
100 ^2+ 61 = 10061, 4* 10061 +1 =40245 = 3*5*2683
100 ^2+ 62 = 10062, 4* 10062 +1 =40249 = 11*3659
100 ^2+ 63 = 10063, 4* 10063 +1 =40253 = 这是一个质数
100 ^2+ 64 = 10064, 4* 10064 +1 =40257 = 3*3*3*3*7*71
100 ^2+ 65 = 10065, 4* 10065 +1 =40261 = 13*19*163
100 ^2+ 66 = 10066, 4* 10066 +1 =40265 = 5*8053
100 ^2+ 67 = 10067, 4* 10067 +1 =40269 = 3*31*433
100 ^2+ 68 = 10068, 4* 10068 +1 =40273 = 17*23*103
100 ^2+ 69 = 10069, 4* 10069 +1 =40277 = 这是一个质数
100 ^2+ 70 = 10070, 4* 10070 +1 =40281 = 3*29*463
100 ^2+ 71 = 10071, 4* 10071 +1 =40285 = 5*7*1151
100 ^2+ 72 = 10072, 4* 10072 +1 =40289 = 这是一个质数
100 ^2+ 73 = 10073, 4* 10073 +1 =40293 = 3*3*11*11*37
100 ^2+ 74 = 10074, 4* 10074 +1 =40297 = 59*683
100 ^2+ 75 = 10075, 4* 10075 +1 =40301 = 191*211
100 ^2+ 76 = 10076, 4* 10076 +1 =40305 = 3*5*2687
100 ^2+ 77 = 10077, 4* 10077 +1 =40309 = 173*233
100 ^2+ 78 = 10078, 4* 10078 +1 =40313 = 7*13*443
100 ^2+ 79 = 10079, 4* 10079 +1 =40317 = 3*89*151
100 ^2+ 80 = 10080, 4* 10080 +1 =40321 = 61*661
100 ^2+ 81 = 10081, 4* 10081 +1 =40325 = 5*5*1613
100 ^2+ 82 = 10082, 4* 10082 +1 =40329 = 3*3*4481
100 ^2+ 83 = 10083, 4* 10083 +1 =40333 = 53*761
100 ^2+ 84 = 10084, 4* 10084 +1 =40337 = 11*19*193
100 ^2+ 85 = 10085, 4* 10085 +1 =40341 = 3*7*17*113
100 ^2+ 86 = 10086, 4* 10086 +1 =40345 = 5*8069
100 ^2+ 87 = 10087, 4* 10087 +1 =40349 = 157*257
100 ^2+ 88 = 10088, 4* 10088 +1 =40353 = 3*13451
100 ^2+ 89 = 10089, 4* 10089 +1 =40357 = 这是一个质数
100 ^2+ 90 = 10090, 4* 10090 +1 =40361 = 这是一个质数
100 ^2+ 91 = 10091, 4* 10091 +1 =40365 = 3*3*3*5*13*23
100 ^2+ 92 = 10092, 4* 10092 +1 =40369 = 7*73*79
100 ^2+ 93 = 10093, 4* 10093 +1 =40373 = 47*859
100 ^2+ 94 = 10094, 4* 10094 +1 =40377 = 3*43*313
100 ^2+ 95 = 10095, 4* 10095 +1 =40381 = 11*3671
100 ^2+ 96 = 10096, 4* 10096 +1 =40385 = 5*41*197
100 ^2+ 97 = 10097, 4* 10097 +1 =40389 = 3*13463
100 ^2+ 98 = 10098, 4* 10098 +1 =40393 = 31*1303
100 ^2+ 99 = 10099, 4* 10099 +1 =40397 = 7*29*199
100 ^2+ 100 = 10100, 4* 10100 +1 =40401 = 3*3*67*67

ysr

Private Sub Command1_Click()
Dim k, n
n = Val(Text1)
k = 1
Do While k <= n
m = Val(n ^ 2 + k)
p = Val(4 * m + 3)
f = fenjieyinzi(Val(p))
s1 = s1 & n & " ^2+ " & k & " = " & m & ", 4* " & m & " +3 =" & p & " = " & f & vbCrLf
k = k + 1
Loop
Text2 = s1

End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""

End Sub

Private Function fenjieyinzi(sa As String) As String
Dim x, a, b, k As String
a = Val(sa)

x = 3
If a <= 1 Or a > Int(a) Then
If a = 1 Then
fenjieyinzi = "它既不是质数，也不是合数"

Else
MsgBox "error"
End If
  
Else

Do While a / 2 = Int(a / 2) And a >= 4
  
If b = 0 Then
fenjieyinzi = fenjieyinzi & "2"
b = 1
Else
fenjieyinzi = fenjieyinzi & "*2"
End If
a = a / 2
k = a
  
Loop

Do While a > 1
Do While x <= Sqr(a)
Do While a / x = Int(a / x) And a >= x * x
  
If b = 0 Then
fenjieyinzi = fenjieyinzi & x
b = 1
Else
fenjieyinzi = fenjieyinzi & "*" & x
End If
a = a / x
Loop
  
x = x + 2
Loop
  
k = a
a = 1
Loop
  
If b = 1 Then
fenjieyinzi = fenjieyinzi & "*" & k
Else
fenjieyinzi = "这是一个质数"
End If
  
End If

End Function

ysr

100 ^2+ 1 = 10001, 4* 10001 +3 =40007 = 11*3637
100 ^2+ 2 = 10002, 4* 10002 +3 =40011 = 3*13337
100 ^2+ 3 = 10003, 4* 10003 +3 =40015 = 5*53*151
100 ^2+ 4 = 10004, 4* 10004 +3 =40019 = 7*5717
100 ^2+ 5 = 10005, 4* 10005 +3 =40023 = 3*3*4447
100 ^2+ 6 = 10006, 4* 10006 +3 =40027 = 13*3079
100 ^2+ 7 = 10007, 4* 10007 +3 =40031 = 这是一个质数
100 ^2+ 8 = 10008, 4* 10008 +3 =40035 = 3*5*17*157
100 ^2+ 9 = 10009, 4* 10009 +3 =40039 = 这是一个质数
100 ^2+ 10 = 10010, 4* 10010 +3 =40043 = 23*1741
100 ^2+ 11 = 10011, 4* 10011 +3 =40047 = 3*7*1907
100 ^2+ 12 = 10012, 4* 10012 +3 =40051 = 11*11*331
100 ^2+ 13 = 10013, 4* 10013 +3 =40055 = 5*8011
100 ^2+ 14 = 10014, 4* 10014 +3 =40059 = 3*3*4451
100 ^2+ 15 = 10015, 4* 10015 +3 =40063 = 这是一个质数
100 ^2+ 16 = 10016, 4* 10016 +3 =40067 = 103*389
100 ^2+ 17 = 10017, 4* 10017 +3 =40071 = 3*19*19*37
100 ^2+ 18 = 10018, 4* 10018 +3 =40075 = 5*5*7*229
100 ^2+ 19 = 10019, 4* 10019 +3 =40079 = 13*3083
100 ^2+ 20 = 10020, 4* 10020 +3 =40083 = 3*31*431
100 ^2+ 21 = 10021, 4* 10021 +3 =40087 = 这是一个质数
100 ^2+ 22 = 10022, 4* 10022 +3 =40091 = 47*853
100 ^2+ 23 = 10023, 4* 10023 +3 =40095 = 3*3*3*3*3*3*5*11
100 ^2+ 24 = 10024, 4* 10024 +3 =40099 = 这是一个质数
100 ^2+ 25 = 10025, 4* 10025 +3 =40103 = 7*17*337
100 ^2+ 26 = 10026, 4* 10026 +3 =40107 = 3*29*461
100 ^2+ 27 = 10027, 4* 10027 +3 =40111 = 这是一个质数
100 ^2+ 28 = 10028, 4* 10028 +3 =40115 = 5*71*113
100 ^2+ 29 = 10029, 4* 10029 +3 =40119 = 3*43*311
100 ^2+ 30 = 10030, 4* 10030 +3 =40123 = 这是一个质数
100 ^2+ 31 = 10031, 4* 10031 +3 =40127 = 这是一个质数
100 ^2+ 32 = 10032, 4* 10032 +3 =40131 = 3*3*7*7*7*13
100 ^2+ 33 = 10033, 4* 10033 +3 =40135 = 5*23*349
100 ^2+ 34 = 10034, 4* 10034 +3 =40139 = 11*41*89
100 ^2+ 35 = 10035, 4* 10035 +3 =40143 = 3*13381
100 ^2+ 36 = 10036, 4* 10036 +3 =40147 = 19*2113
100 ^2+ 37 = 10037, 4* 10037 +3 =40151 = 这是一个质数
100 ^2+ 38 = 10038, 4* 10038 +3 =40155 = 3*5*2677
100 ^2+ 39 = 10039, 4* 10039 +3 =40159 = 7*5737
100 ^2+ 40 = 10040, 4* 10040 +3 =40163 = 这是一个质数
100 ^2+ 41 = 10041, 4* 10041 +3 =40167 = 3*3*4463
100 ^2+ 42 = 10042, 4* 10042 +3 =40171 = 17*17*139
100 ^2+ 43 = 10043, 4* 10043 +3 =40175 = 5*5*1607
100 ^2+ 44 = 10044, 4* 10044 +3 =40179 = 3*59*227
100 ^2+ 45 = 10045, 4* 10045 +3 =40183 = 11*13*281
100 ^2+ 46 = 10046, 4* 10046 +3 =40187 = 7*5741
100 ^2+ 47 = 10047, 4* 10047 +3 =40191 = 3*13397
100 ^2+ 48 = 10048, 4* 10048 +3 =40195 = 5*8039
100 ^2+ 49 = 10049, 4* 10049 +3 =40199 = 61*659
100 ^2+ 50 = 10050, 4* 10050 +3 =40203 = 3*3*3*1489
100 ^2+ 51 = 10051, 4* 10051 +3 =40207 = 31*1297
100 ^2+ 52 = 10052, 4* 10052 +3 =40211 = 79*509
100 ^2+ 53 = 10053, 4* 10053 +3 =40215 = 3*5*7*383
100 ^2+ 54 = 10054, 4* 10054 +3 =40219 = 37*1087
100 ^2+ 55 = 10055, 4* 10055 +3 =40223 = 19*29*73
100 ^2+ 56 = 10056, 4* 10056 +3 =40227 = 3*11*23*53
100 ^2+ 57 = 10057, 4* 10057 +3 =40231 = 这是一个质数
100 ^2+ 58 = 10058, 4* 10058 +3 =40235 = 5*13*619
100 ^2+ 59 = 10059, 4* 10059 +3 =40239 = 3*3*17*263
100 ^2+ 60 = 10060, 4* 10060 +3 =40243 = 7*5749
100 ^2+ 61 = 10061, 4* 10061 +3 =40247 = 167*241
100 ^2+ 62 = 10062, 4* 10062 +3 =40251 = 3*13417
100 ^2+ 63 = 10063, 4* 10063 +3 =40255 = 5*83*97
100 ^2+ 64 = 10064, 4* 10064 +3 =40259 = 127*317
100 ^2+ 65 = 10065, 4* 10065 +3 =40263 = 3*13421
100 ^2+ 66 = 10066, 4* 10066 +3 =40267 = 67*601
100 ^2+ 67 = 10067, 4* 10067 +3 =40271 = 7*11*523
100 ^2+ 68 = 10068, 4* 10068 +3 =40275 = 3*3*5*5*179
100 ^2+ 69 = 10069, 4* 10069 +3 =40279 = 47*857
100 ^2+ 70 = 10070, 4* 10070 +3 =40283 = 这是一个质数
100 ^2+ 71 = 10071, 4* 10071 +3 =40287 = 3*13*1033
100 ^2+ 72 = 10072, 4* 10072 +3 =40291 = 43*937
100 ^2+ 73 = 10073, 4* 10073 +3 =40295 = 5*8059
100 ^2+ 74 = 10074, 4* 10074 +3 =40299 = 3*7*19*101
100 ^2+ 75 = 10075, 4* 10075 +3 =40303 = 41*983
100 ^2+ 76 = 10076, 4* 10076 +3 =40307 = 17*2371
100 ^2+ 77 = 10077, 4* 10077 +3 =40311 = 3*3*3*1493
100 ^2+ 78 = 10078, 4* 10078 +3 =40315 = 5*11*733
100 ^2+ 79 = 10079, 4* 10079 +3 =40319 = 23*1753
100 ^2+ 80 = 10080, 4* 10080 +3 =40323 = 3*13441
100 ^2+ 81 = 10081, 4* 10081 +3 =40327 = 7*7*823
100 ^2+ 82 = 10082, 4* 10082 +3 =40331 = 31*1301
100 ^2+ 83 = 10083, 4* 10083 +3 =40335 = 3*5*2689
100 ^2+ 84 = 10084, 4* 10084 +3 =40339 = 13*29*107
100 ^2+ 85 = 10085, 4* 10085 +3 =40343 = 这是一个质数
100 ^2+ 86 = 10086, 4* 10086 +3 =40347 = 3*3*4483
100 ^2+ 87 = 10087, 4* 10087 +3 =40351 = 这是一个质数
100 ^2+ 88 = 10088, 4* 10088 +3 =40355 = 5*7*1153
100 ^2+ 89 = 10089, 4* 10089 +3 =40359 = 3*11*1223
100 ^2+ 90 = 10090, 4* 10090 +3 =40363 = 181*223
100 ^2+ 91 = 10091, 4* 10091 +3 =40367 = 37*1091
100 ^2+ 92 = 10092, 4* 10092 +3 =40371 = 3*13457
100 ^2+ 93 = 10093, 4* 10093 +3 =40375 = 5*5*5*17*19
100 ^2+ 94 = 10094, 4* 10094 +3 =40379 = 149*271
100 ^2+ 95 = 10095, 4* 10095 +3 =40383 = 3*3*7*641
100 ^2+ 96 = 10096, 4* 10096 +3 =40387 = 这是一个质数
100 ^2+ 97 = 10097, 4* 10097 +3 =40391 = 13*13*239
100 ^2+ 98 = 10098, 4* 10098 +3 =40395 = 3*5*2693
100 ^2+ 99 = 10099, 4* 10099 +3 =40399 = 71*569
100 ^2+ 100 = 10100, 4* 10100 +3 =40403 = 11*3673

ysr

对于*4+3型的合数，n=15015时，也不行因为15015 ^2+ 14 = 225450239, 4* 225450239 +3 =901800959 = 这是一个质数

AI说n=304250263527210的时候，满足条件，数值太大我的程序无法验证了

这是不对的，根据《数论探秘》中的定理10和定理11，这个区间内必然有不止一个素数，定理10和定理11的区间都落在杰波夫区间内了，而且随着n的增大，区间不止一个而且还是不断增多的。
这么大的数，该区间内会有很多素数，甚至可能是成千上万个素数。

ysr

2 ^2+ 1 = 5, 4* 5 +3 =23 = 这是一个质数
2 ^2+ 2 = 6, 4* 6 +3 =27 = 3*3*3