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 February 15th, 2011, 01:51 PM #1 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 n!*(n+1)! - 1 composite? Hello everybody, n>5 I have found that for n=6 to 30 : n!*(n+1)! - 1 is always composite. I want to know if that is true or not with n>30
 February 15th, 2011, 04:11 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: n!*(n+1)! - 1 composite? It's prime for 76, 166, 344, 394 and so forth. There should be infinitely many, though for precise asymptotics I'd need to do some calculations.
 February 15th, 2011, 04:56 PM #3 Senior Member   Joined: Oct 2008 Posts: 215 Thanks: 0 Re: n!*(n+1)! - 1 composite? For a random number near to n!*(n+1)!-1, the probability that it is a prime is around $\frac{1}{\log(n!(n+1)!)}\tilde= \frac{1}{2n\log(n)}$ For any random number near n, the probability that it is a prime is around $\frac{1}{\log(n)}$ Given n!*(n+1)!-1 has no prime factor less than n, this should be a similar probability as a number less than n to be prime, we could estimate that the probability that n!(n+1)!-1 to be prime is around $\frac{1}{2n}$. Although the probability is low, it supports CRG's claim that there're an infinity of numbers.
 February 16th, 2011, 01:27 AM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: n!*(n+1)! - 1 composite? Right. Even without the Mertens adjustment, though, we expect infinitely many -- just not as dense. I don't find any more terms to 3265.
 February 16th, 2011, 02:29 AM #5 Senior Member   Joined: Dec 2007 Posts: 687 Thanks: 47 Re: n!*(n+1)! - 1 composite? Hi bogauss, for instance, this polynomial $x^2+x+1$ gives prime numbers from (x=1,1000) at the following 189 values: {for(x=1,1000,if(isprime(x^2+x+1),print(x)))} Code: 1 2 3 5 6 8 12 14 15 17 20 21 24 27 33 38 41 50 54 57 59 62 66 69 71 75 77 78 80 89 90 99 101 105 110 111 117 119 131 138 141 143 147 150 153 155 161 162 164 167 168 173 176 188 189 192 194 203 206 209 215 218 231 236 245 246 266 272 278 279 287 288 290 293 309 314 329 332 336 342 344 348 351 357 369 378 381 383 392 395 398 402 404 405 414 416 426 434 435 447 453 455 456 476 489 495 500 512 518 525 530 531 533 537 540 551 554 560 566 567 572 579 582 584 603 605 609 612 621 624 626 635 642 644 668 671 677 686 696 701 720 726 728 735 743 747 755 761 762 768 773 782 785 792 798 801 812 818 819 825 827 836 839 846 855 857 860 864 875 878 890 894 897 899 911 915 918 920 927 950 959 960 969 974 981 987 990 992 993 and this one $x^2+x-1$, switching the signal of 1, gives prime numbers from (x=1,1000) at the following 312 values: {for(x=1,1000,if(isprime(x^2+x-1),print(x)))} Code: 2 3 4 5 6 8 9 10 11 13 15 16 19 20 21 24 26 28 30 31 35 38 39 41 44 45 46 48 50 53 54 55 56 59 60 64 65 66 68 70 76 83 85 86 89 93 94 96 100 101 103 114 115 120 125 126 130 131 134 138 140 141 144 145 148 149 153 154 155 158 159 160 163 164 169 171 174 176 180 181 184 186 188 191 193 195 196 199 203 206 209 215 218 219 220 225 230 231 233 236 240 241 244 246 248 258 259 263 264 265 268 281 285 288 290 294 296 298 301 303 305 306 309 314 319 323 330 331 335 339 343 349 350 351 354 358 360 361 364 373 374 378 380 385 386 391 393 395 396 401 405 408 418 419 420 423 428 431 433 445 448 449 450 453 455 456 461 463 468 471 473 474 475 478 481 486 490 494 495 496 499 500 501 504 506 510 511 515 518 519 528 530 541 544 548 549 550 551 560 561 569 571 573 578 581 588 589 591 594 595 596 598 604 606 610 616 618 624 625 629 633 639 640 644 646 648 651 654 655 659 668 670 673 676 683 686 691 693 695 699 705 706 708 709 715 716 720 724 728 739 743 746 750 753 754 761 763 765 769 776 779 780 781 786 794 798 803 804 809 814 815 816 825 833 838 845 848 856 863 873 881 884 885 889 890 901 906 915 921 923 925 930 939 941 944 946 950 955 956 963 965 966 968 974 976 981 984 989 994 996 998 1000 now the curious thing, could someone explain why the difference of ~ 64%? something to do with polynomial cyclotomicness I guess (do this word exists???) ps: for x = 1.000.000 is ~ 63,2%
 February 16th, 2011, 04:15 AM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: n!*(n+1)! - 1 composite? x^2 + x + 1 is divisible by 3 1/3 of the time, while x^2 + x - 1 is never divisible by 3. That accounts for 33% of the difference. If you find residues over all the primes you'd presumably find the required number.
February 16th, 2011, 05:43 AM   #7
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Re: n!*(n+1)! - 1 composite?

Quote:
 Originally Posted by CRGreathouse x^2 + x + 1 is divisible by 3 1/3 of the time, while x^2 + x - 1 is never divisible by 3. That accounts for 33% of the difference. If you find residues over all the primes you'd presumably find the required number.
I understand, you're correct.

I was curious about cyclotomic polynomials $x^{p-1}+x^{p-2}+...+x+1$ which are clearly divisible by p 1/p of the time and why, lets say, $x^p+x^{p-1}+x^{p-2}+...+x+1=composite$, or any p such that p+1 is not a prime number, and how this behave changing the signals arbitrarily. That is, some cyclotomic polynomials yields composite numbers all the time, and I suspect it is because not all polynomials of the form $x^n\pm x^{n-1}\pm x^{n-2}\pm ...\pm x\pm 1$ are irreductible by eiseinstein's criterium... am I correct?

February 16th, 2011, 09:30 AM   #8
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Re: n!*(n+1)! - 1 composite?

Quote:
 Originally Posted by al-mahed I was curious about cyclotomic polynomials $x^{p-1}+x^{p-2}+...+x+1$ which are clearly divisible by p 1/p of the time
Not a big deal; that's just one prime, right?

Quote:
 Originally Posted by al-mahed signals
I don't know what you mean when you write this.

Quote:
 Originally Posted by al-mahed That is, some cyclotomic polynomials yields composite numbers all the time
What, you mean like cyclic polynomials at composite values like $\Phi_{mn}$? Sure, those are reducible polynomials.

 February 16th, 2011, 01:34 PM #9 Senior Member   Joined: Dec 2007 Posts: 687 Thanks: 47 Re: n!*(n+1)! - 1 composite? Sorry, my message was very confusing, let me try to be concise and to ask the correct question: is there polynomials that are not reductible but even so yields only composite numbers? I know there is no polynomial that yields only primes, so I was curious about it, just a mere curiosity. I suspect there is something trivial about it, but I'm not "getting there". So I was thinking about cyclotomic polynomials because they seems to be directly related to what I'm wondering.
February 16th, 2011, 01:57 PM   #10
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Re: n!*(n+1)! - 1 composite?

Quote:
 Originally Posted by al-mahed Sorry, my message was very confusing, let me try to be concise and to ask the correct question: is there polynomials that are not reductible but even so yields only composite numbers?
Good question!

The answer is yes. One example is x^2 + x + 2 = x(x +1) + 2. Both summands are even, so its value is even, and for positive x, x^2 + x + 2 > 2. (If you want to get the result for all integers, use x^4 + 6*x^3 + 11*x^2 + 6*x + 4 instead, but the real interest is in the infinite part of the behavior, where you can ignore finite numbers of primes.)

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