World!OfNumbers |
WON plate 197 | |
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[ Let me start with a simple example to illustrate things
Suppose the number 5 is our next number to investigate
( '^^' is symbol for concatenation ) [5][1^^1] = [5][1] = 51 = 3 x 17 = composite
So if this is our next record delayed prime
There are cases whereby appending any number of 1's, 3's, 7's or 9's
For the sake of this wonplate it is not about enlisting all factors
Already number [12] set a milestone as it needed 136 1's
Click on the header titles to open the related worksheets.
Here is an elaborated example of how the factorization of
The first 21 factorizations produce the next list [38][1^^1] = 3 x 127 [38][1^^2] = 37 x 103 [38][1^^3] = 23 x 1657 [38][1^^4] = 3 x 127037 [38][1^^5] = 17 x 37 x 73 x 83 [38][1^^6] = 233 x 163567 [38][1^^7] = 3^2 x 42345679 [38][1^^8] = 37 x 113 x 613 x 1487 [38][1^^9] = 31 x 2333 x 526957 [38][1^^10] = 3 x 2399 x 52954163 [38][1^^11] = 37 x 103003003003 [38][1^^12] = 23333 x 1633356667 [38][1^^13] = 3 x 73 x 1740233384069 [38][1^^14] = 37 x 2287 x 45038479669 [38][1^^15] = 353 x 661 x 163333566667 [38][1^^16] = 3^2 x 131 x 323249458109509 [38][1^^17] = 37 x 114346289 x 900798827 [38][1^^18] = 19 x 227 x 541 x 2857 x 5716953331 [38][1^^19] = 3 x 879449 x 1140233 x 126685261 [38][1^^20] = 37 x 393380951 x 261840342653 [38][1^^21] = 17 x 73 x 1372549 x 22374429543379 or {1 +3} for short are all divisible by 3, so these can be put aside. Idem dito for {2, +3} where all numbers can be divided by 37.
If I shift these to be ignored cases you see what is left over. [38][1^^1] = 3 x 127 [38][1^^2] = 37 x 103 [38][1^^3] = 23 x 1657 [38][1^^4] = 3 x 127037 [38][1^^5] = 17 x 37 x 73 x 83 [38][1^^6] = 233 x 163567 [38][1^^7] = 3^2 x 42345679 [38][1^^8] = 37 x 113 x 613 x 1487 [38][1^^9] = 31 x 2333 x 526957 [38][1^^10] = 3 x 2399 x 52954163 [38][1^^11] = 37 x 103003003003 [38][1^^12] = 23333 x 1633356667 [38][1^^13] = 3 x 73 x 1740233384069 [38][1^^14] = 37 x 2287 x 45038479669 [38][1^^15] = 353 x 661 x 163333566667 [38][1^^16] = 3^2 x 131 x 323249458109509 [38][1^^17] = 37 x 114346289 x 900798827 [38][1^^18] = 19 x 227 x 541 x 2857 x 5716953331 [38][1^^19] = 3 x 879449 x 1140233 x 126685261 [38][1^^20] = 37 x 393380951 x 261840342653 [38][1^^21] = 17 x 73 x 1372549 x 22374429543379
Spotting [38][1^^3], [38][1^^6], [38][1^^12] one sees that we are
Take e.g. [38][1^^21] = 17 x 73 x 1372549 x 22374429543379
The general formula for [38][1^^21]{3 +3} becomes Here is the extracted list of genuine semiprimes for [38][1^^21]{3 +3} [38][1^^3] ➜ m = 1 ➜ 23 x 1657 [38][1^^6] ➜ m = 2 ➜ 233 x 163567 [38][1^^12] ➜ m = 4 ➜ 23333 x 1633356667 [38][1^^66] ➜ m = 22 ➜ [2][3^^22] x [16][3^^21][5][6^^21][7] or 23333333333333333333333 x 1633333333333333333333356666666666666666666667
Can you find more of these semiprimes ?
Note : a preliminary search revealed that the next semiprime
Xinyao Chen investigated this topic | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

A000197 Prime Curios! Prime Puzzle Wikipedia 197 Le nombre 197 |

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