Mersenne's on Ulams spiral

2002-11-04, 05:10

I was messing with the prime number spiral, and decided to try odd numbers instead.

To my amazment odd mersenne numbers, fall onto one strip.
In the form of the polynomial 8(n^2) -1. and n is power of two.

The numbers that interest me here, are those of the above form that are not mersenne, but are prime. The study of their density, could be used for finding mersenne primes? :D

toferc · 2002-11-04, 12:44

Quote:

Originally Posted by TTn

... In the form of the polynomial 8(n^2) -1. and n is power of two.

This form can be rewritten as 2^k - 1, where k is odd and greater than one, thus these numbers form a subset of the Mersenne numbers. It is known that 2^k - 1 is prime only if k is prime, therefore all Mersenne primes other than 3 are in this sequence, and they are the only primes in this sequence.

2002-11-04, 21:56

Obviously you weren't listening carefully.
2^k -1 is a specific form.

The densitiy of primes, in the subset form 8(n^2) -1
is the real question, which in turn, effects the density of primes 2^k -1.

2002-11-04, 22:14

Clarification, ALL values N.
The densitiy of primes, in the subset form 8(n^2) -1

Also the densitity of primes in form 8(n^2) +1 /3
so the probability that they may both prime can be analyzed, in respect to the new mersenne conjecture.

Deamiter · 2002-11-05, 02:22

it seems to me that 8(n^2)-1 is a subset of 2^k-1 as long as 'n' is a power of two like you said. That's because 8=2^3 so your form is really (2^3)*(n^2)-1, but since n is always a power of two, you can always simplify the number into the form 2^k-1... I hope you can see why this is... as any (power of two)^2 can always be written in the form 2^n where n will always be even. so your numbers are really in the form 2^(n+3)-1...

and now I'm getting tired, so if that made little sense, wait for one of the GIMPS gurus to respond with a better explanation.

2002-11-05, 10:54

"seems to me that 8(n^2)-1 is a subset of 2^k-1 as long as 'n' is a power of two like you said. "

Stop ! then I go on to say, the interesting numbers are in the above form, BUT are NOT mersenne.

This means I am speaking of all other n, not producing a mersenne number(odd specifically), but such that produce a prime number.

The density could be manipulated, to analyze that of mersenne primes.

If we were talking about the same thing, then there would be primes like,
2^k -1 = 199, where k is not an integer.

I hope this clears it up.

ewmayer · 2002-11-05, 17:38

What you've observed is simply a "rich vein" of primes of a certain
form. The folks searching for huge Proth primes (p = k*2^n + 1) also
look for such veins (in their case, k's that yield a high percentage
of primes) to maximize their chances of finding one.

In your case, it shouldn't be surprising that there are many primes
of the form N = 8*n^2 - 1. If we write this as 2^3*n^2 - 1, we see that
unless n = 2^j and 2*j+3 is composite (i.e. a composite-exponent
Mersenne), the fact that the power on the 2 and the n are relatively
prime means we can't write N as a^k - b^k (in your case b = 1), which
rules out any easy algebraic factors. That significantly boosts the
odds of N being prime. It's also quite likely (I'll leave you to
investigate this) that N of this form, if they are composite, must
have their factors of a certain algebraic form. That would again
increase the prime density, by eliminating most random odd numbers
as candidate factors.

Yes, the Mersennes are a special case of these numbers, but it seems
doubtful that fact will help us find more Mersennes - those are a
special case unto themselves, we know a lot about their properties
and even have a fast deterministic primality test for them. We also
know they occur (AFAWK) for random prime exponents of 2, and that
their density thins out fairly predictably as they get bigger.

FYI, here are all the n <= 10^4 for which 2^3*n^2 - 1 is prime.
You see the general thinning-out trend, with a few hiccups (e.g. the
n = 5000-6000 and 8000-9000 columns are longer than the ones that
precede them), but one sees such behavior for most any reasonably
dense sequence of primes one wishes to investigate:

[code:1]
1 1003 2004 3013 4006 5012 6001 7003 8008 9016
2 1004 2012 3014 4008 5014 6002 7007 8012 9018
3 1008 2020 3015 4011 5015 6004 7018 8013 9027
4 1011 2021 3017 4037 5017 6013 7023 8018 9033
5 1018 2023 3022 4046 5033 6017 7037 8029 9040
9 1026 2026 3027 4048 5035 6020 7038 8032 9041
11 1032 2034 3031 4057 5036 6024 7049 8039 9042
12 1033 2035 3035 4058 5052 6027 7051 8040 9056
14 1034 2051 3038 4067 5058 6029 7056 8046 9065
17 1036 2053 3049 4071 5065 6034 7061 8052 9074
18 1038 2068 3057 4074 5068 6037 7063 8060 9077
19 1039 2072 3061 4076 5075 6044 7067 8075 9082
21 1041 2083 3062 4077 5085 6052 7072 8078 9086
23 1045 2090 3068 4081 5092 6055 7094 8081 9089
25 1048 2095 3077 4084 5098 6060 7098 8087 9090
26 1050 2103 3080 4088 5103 6072 7102 8101 9093
28 1053 2104 3096 4091 5106 6080 7105 8104 9097
31 1055 2107 3105 4095 5112 6083 7117 8109 9100
32 1066 2112 3113 4111 5120 6086 7119 8113 9109
38 1069 2114 3115 4113 5121 6087 7124 8122 9110
40 1075 2117 3117 4116 5128 6094 7142 8127 9114
46 1080 2119 3119 4118 5135 6101 7143 8134 9118
49 1082 2124 3125 4123 5143 6107 7144 8137 9121
51 1085 2128 3140 4128 5150 6114 7152 8139 9130
54 1092 2139 3143 4132 5152 6116 7156 8144 9131
56 1094 2140 3145 4137 5154 6118 7157 8145 9132
59 1099 2142 3153 4139 5170 6121 7158 8160 9144
63 1102 2145 3154 4142 5183 6122 7165 8174 9145
66 1104 2147 3157 4144 5184 6129 7170 8176 9146
67 1106 2151 3161 4148 5190 6135 7171 8185 9166
70 1122 2153 3168 4154 5194 6137 7178 8197 9179
77 1125 2154 3175 4160 5197 6141 7179 8202 9182
79 1127 2158 3178 4161 5199 6142 7180 8207 9189
80 1131 2160 3180 4163 5204 6146 7189 8214 9203
82 1136 2165 3187 4177 5220 6150 7199 8229 9210
86 1137 2175 3201 4191 5231 6155 7200 8232 9223
89 1148 2180 3202 4198 5233 6158 7203 8234 9235
93 1150 2191 3209 4200 5240 6160 7212 8242 9242
94 1152 2196 3210 4203 5247 6162 7214 8253 9244
96 1153 2210 3230 4204 5252 6165 7220 8265 9245
98 1158 2214 3232 4207 5255 6170 7224 8267 9250
100 1159 2226 3234 4210 5261 6186 7227 8270 9259
102 1164 2229 3241 4211 5273 6190 7231 8271 9268
103 1169 2236 3255 4228 5275 6193 7234 8274 9278
107 1172 2240 3262 4230 5281 6199 7236 8277 9282
110 1174 2242 3264 4237 5282 6205 7240 8278 9287
114 1178 2243 3266 4247 5285 6211 7241 8281 9291
116 1179 2244 3278 4251 5287 6216 7254 8283 9293
119 1181 2252 3279 4252 5296 6218 7255 8284 9303
121 1187 2256 3283 4266 5302 6220 7259 8292 9305
124 1188 2257 3304 4272 5309 6223 7262 8298 9315
128 1190 2259 3313 4281 5310 6234 7264 8302 9317
133 1192 2264 3314 4282 5313 6235 7270 8313 9319
135 1193 2272 3320 4286 5317 6237 7276 8316 9324
137 1201 2273 3321 4293 5332 6239 7277 8325 9327
140 1213 2277 3323 4295 5341 6265 7291 8328 9328
144 1215 2278 3329 4298 5346 6268 7296 8339 9335
147 1218 2279 3332 4300 5353 6269 7319 8344 9349
150 1221 2280 3341 4301 5355 6275 7324 8346 9362
152 1223 2286 3344 4303 5357 6276 7326 8347 9366
156 1225 2289 3348 4305 5360 6289 7338 8348 9370
161 1227 2291 3358 4309 5371 6293 7340 8355 9378
166 1228 2299 3362 4316 5373 6302 7357 8361 9382
173 1229 2312 3364 4321 5376 6304 7359 8362 9385
182 1239 2315 3367 4326 5386 6307 7360 8384 9389
186 1241 2317 3371 4329 5388 6310 7361 8389 9396
187 1243 2320 3379 4330 5390 6312 7375 8398 9404
189 1250 2321 3384 4333 5401 6319 7378 8407 9405
192 1253 2324 3392 4344 5402 6337 7382 8412 9413
193 1260 2326 3399 4349 5409 6338 7383 8417 9417
201 1263 2333 3414 4350 5411 6339 7392 8418 9439
203 1264 2334 3416 4351 5414 6347 7394 8419 9447
205 1267 2335 3418 4358 5436 6352 7395 8424 9455
207 1271 2338 3428 4370 5437 6370 7396 8426 9464
208 1276 2342 3430 4371 5439 6372 7397 8428 9467
213 1283 2357 3432 4375 5441 6375 7406 8432 9468
220 1288 2359 3437 4378 5443 6379 7408 8433 9471
221 1292 2363 3446 4380 5446 6386 7409 8435 9473
222 1293 2366 3449 4382 5448 6389 7410 8437 9475
224 1298 2369 3453 4385 5453 6394 7418 8438 9478
226 1307 2375 3456 4389 5455 6395 7429 8444 9481
229 1309 2377 3463 4392 5463 6396 7430 8447 9489
233 1312 2378 3472 4401 5474 6398 7437 8449 9490
234 1314 2382 3477 4406 5476 6405 7443 8451 9499
235 1323 2391 3483 4414 5478 6409 7462 8453 9508
236 1325 2396 3484 4419 5486 6414 7464 8458 9522
241 1330 2397 3486 4420 5495 6423 7465 8467 9523
243 1337 2403 3490 4424 5497 6437 7466 8474 9532
249 1339 2406 3493 4429 5506 6438 7474 8482 9541
254 1342 2410 3497 4433 5521 6442 7479 8489 9553
256 1348 2429 3504 4436 5525 6444 7481 8498 9560
257 1349 2432 3507 4438 5526 6445 7488 8500 9565
263 1356 2439 3511 4450 5530 6451 7494 8509 9566
264 1360 2443 3517 4454 5533 6459 7495 8512 9567
266 1363 2446 3518 4457 5539 6463 7506 8515 9574
271 1368 2450 3524 4473 5544 6485 7508 8530 9579
273 1369 2452 3528 4480 5546 6492 7513 8537 9585
276 1374 2457 3530 4485 5548 6498 7516 8545 9586
278 1375 2459 3531 4491 5553 6499 7518 8551 9593
285 1376 2461 3538 4496 5560 6506 7520 8559 9594
290 1377 2464 3542 4499 5561 6507 7522 8565 9597
291 1382 2467 3551 4508 5562 6515 7536 8568 9599
301 1389 2473 3556 4517 5565 6517 7548 8571 9604
303 1414 2483 3565 4531 5568 6522 7556 8577 9606
304 1425 2485 3566 4533 5572 6524 7563 8579 9609
305 1426 2487 3570 4548 5579 6527 7565 8580 9613
310 1430 2494 3574 4552 5597 6536 7569 8582 9618
311 1431 2499 3586 4553 5602 6545 7585 8593 9622
317 1437 2502 3588 4567 5607 6559 7588 8596 9634
318 1439 2504 3589 4571 5614 6573 7598 8599 9635
325 1449 2510 3591 4587 5618 6582 7599 8603 9650
326 1453 2511 3596 4588 5624 6583 7607 8605 9651
327 1460 2515 3615 4589 5625 6584 7621 8608 9657
345 1463 2516 3616 4602 5632 6591 7625 8614 9662
348 1475 2520 3617 4608 5635 6598 7633 8615 9664
353 1477 2524 3622 4613 5638 6604 7644 8628 9665
354 1480 2529 3626 4615 5645 6611 7653 8635 9671
355 1481 2530 3630 4620 5646 6612 7658 8650 9678
359 1487 2531 3635 4629 5647 6615 7662 8652 9684
360 1493 2538 3638 4632 5660 6619 7667 8654 9688
368 1496 2541 3640 4636 5661 6624 7669 8656 9693
371 1498 2552 3644 4637 5670 6629 7672 8661 9702
378 1505 2564 3649 4638 5672 6643 7676 8669 9706
382 1508 2566 3659 4650 5673 6646 7679 8670 9709
383 1512 2576 3661 4655 5679 6652 7681 8675 9716
387 1516 2578 3663 4657 5681 6661 7682 8676 9720
390 1524 2580 3668 4658 5684 6662 7684 8683 9725
392 1526 2586 3670 4659 5689 6668 7688 8690 9726
395 1530 2587 3671 4660 5694 6669 7696 8691 9735
396 1533 2590 3672 4666 5695 6675 7697 8699 9737
402 1536 2593 3677 4669 5696 6680 7700 8704 9740
408 1538 2599 3678 4671 5708 6681 7704 8708 9741
409 1543 2604 3685 4672 5714 6682 7705 8715 9744
410 1544 2607 3692 4673 5717 6683 7716 8717 9747
420 1545 2616 3693 4681 5728 6685 7719 8720 9758
424 1547 2621 3694 4686 5730 6687 7723 8722 9761
425 1550 2623 3701 4688 5731 6692 7732 8725 9762
427 1551 2632 3705 4704 5735 6696 7738 8741 9770
431 1556 2634 3707 4706 5738 6699 7739 8746 9776
438 1563 2643 3717 4711 5740 6702 7740 8750 9777
441 1578 2646 3720 4723 5742 6706 7761 8755 9779
448 1584 2648 3727 4727 5750 6715 7770 8760 9784
455 1600 2651 3728 4729 5751 6741 7772 8764 9797
458 1607 2655 3729 4730 5758 6744 7774 8766 9807
459 1612 2656 3734 4732 5761 6753 7777 8769 9809
462 1620 2657 3736 4742 5772 6766 7794 8774 9810
465 1621 2658 3752 4746 5785 6767 7800 8776 9821
472 1624 2667 3755 4748 5798 6774 7801 8781 9824
474 1628 2670 3757 4749 5806 6785 7815 8788 9832
479 1633 2685 3771 4751 5827 6795 7817 8790 9844
481 1643 2690 3773 4755 5834 6806 7829 8792 9852
485 1648 2691 3775 4758 5840 6809 7840 8803 9854
492 1652 2692 3777 4765 5842 6820 7842 8804 9858
495 1661 2697 3782 4774 5843 6823 7854 8811 9881
497 1664 2702 3785 4778 5845 6828 7856 8815 9882
502 1666 2705 3787 4783 5847 6829 7858 8820 9886
508 1670 2706 3790 4785 5852 6836 7863 8823 9888
509 1678 2711 3792 4786 5856 6839 7868 8829 9893
514 1683 2718 3803 4802 5863 6843 7871 8831 9896
516 1685 2719 3805 4806 5864 6846 7875 8832 9905
521 1698 2723 3808 4809 5866 6848 7877 8844 9908
523 1699 2735 3820 4811 5873 6867 7879 8851 9914
532 1704 2740 3822 4813 5876 6876 7887 8852 9915
541 1706 2746 3826 4819 5884 6885 7899 8853 9924
542 1711 2748 3827 4839 5894 6888 7910 8857 9926
544 1712 2751 3829 4840 5911 6897 7917 8860 9933
548 1713 2754 3833 4842 5922 6898 7924 8866 9942
549 1720 2756 3854 4848 5924 6900 7927 8869 9944
553 1722 2770 3867 4858 5931 6913 7935 8888 9951
555 1731 2793 3868 4861 5955 6921 7940 8897 9964
563 1733 2796 3875 4870 5968 6925 7941 8902 9987
564 1740 2797 3876 4874 5988 6928 7943 8906 9991
567 1743 2802 3881 4875 5990 6941 7945 8907 9992
569 1745 2804 3885 4877 5995 6944 7954 8911 9999
570 1747 2817 3887 4881 5997 6948 7961 8913 10000
572 1752 2818 3892 4882 5999 6949 7962 8923
577 1753 2821 3895 4884 6951 7964 8927
578 1754 2824 3899 4891 6956 7975 8936
579 1759 2825 3906 4905 6968 7978 8941
581 1762 2828 3910 4907 6979 7985 8942
583 1768 2831 3915 4909 6982 7998 8948
584 1773 2840 3916 4917 6984 8956
593 1788 2844 3922 4918 6986 8979
599 1789 2851 3936 4921 6996 8981
600 1790 2860 3943 4924 8993
611 1794 2867 3944 4932 8995
613 1797 2870 3945 4933 8998
616 1803 2873 3946 4947
625 1804 2874 3952 4963
640 1811 2875 3953 4966
641 1816 2879 3957 4981
644 1820 2888 3959 4982
646 1822 2889 3964 4987
663 1825 2893 3974
667 1830 2895 3976
669 1834 2896 3981
672 1837 2902 3987
675 1845 2909 3993
676 1850 2929 3999
686 1852 2944
688 1855 2949
693 1857 2950
698 1859 2959
700 1864 2963
705 1865 2964
709 1867 2973
712 1872 2977
714 1878 2986
716 1887 2989
717 1900 2993
718 1904 2996
725 1922 3000
726 1929
730 1930
735 1935
737 1944
739 1946
745 1956
747 1957
749 1958
751 1970
753 1976
754 1977
760 1978
767 1981
770 1983
774 1990
779
781
787
793
796
802
805
816
822
824
829
833
845
850
851
865
871
872
873
879
882
887
892
898
899
900
905
907
910
914
915
919
922
935
938
947
954
961
964
968
971
975
983
984
985
987
998
[/code:1]

Re. your comment about N = 2^k - 1 for non-integer k: if I allow k to
be irrational, I can write any positive integer I want this way. So what?

2002-11-06, 02:41

Thanks for the list!
I doubt, the doubt.
The comment 2^k-1 was to verify that we were not on the same page.

2002-11-04, 05:10	#1
TTn 2·37·131 Posts	Mersenne's on Ulams spiral I was messing with the prime number spiral, and decided to try odd numbers instead. To my amazment odd mersenne numbers, fall onto one strip. In the form of the polynomial 8(n^2) -1. and n is power of two. The numbers that interest me here, are those of the above form that are not mersenne, but are prime. The study of their density, could be used for finding mersenne primes? :D

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2002-11-04, 21:56	#3
TTn 5530₁₀ Posts	Obviously you weren't listening carefully. 2^k -1 is a specific form. The densitiy of primes, in the subset form 8(n^2) -1 is the real question, which in turn, effects the density of primes 2^k -1.

2002-11-04, 22:14	#4
TTn 2520₁₀ Posts	Clarification, ALL values N. The densitiy of primes, in the subset form 8(n^2) -1 Also the densitity of primes in form 8(n^2) +1 /3 so the probability that they may both prime can be analyzed, in respect to the new mersenne conjecture.

2002-11-05, 02:22	#5
Deamiter Sep 2002 3²×13 Posts	it seems to me that 8(n^2)-1 is a subset of 2^k-1 as long as 'n' is a power of two like you said. That's because 8=2^3 so your form is really (2^3)*(n^2)-1, but since n is always a power of two, you can always simplify the number into the form 2^k-1... I hope you can see why this is... as any (power of two)^2 can always be written in the form 2^n where n will always be even. so your numbers are really in the form 2^(n+3)-1... and now I'm getting tired, so if that made little sense, wait for one of the GIMPS gurus to respond with a better explanation.

2002-11-05, 10:54	#6
TTn 29×239 Posts	"seems to me that 8(n^2)-1 is a subset of 2^k-1 as long as 'n' is a power of two like you said. " Stop ! then I go on to say, the interesting numbers are in the above form, BUT are NOT mersenne. This means I am speaking of all other n, not producing a mersenne number(odd specifically), but such that produce a prime number. The density could be manipulated, to analyze that of mersenne primes. If we were talking about the same thing, then there would be primes like, 2^k -1 = 199, where k is not an integer. I hope this clears it up.

2002-11-05, 17:38	#7
ewmayer ∂²ω=0 Sep 2002 República de California 5×2,351 Posts	What you've observed is simply a "rich vein" of primes of a certain form. The folks searching for huge Proth primes (p = k2^n + 1) also look for such veins (in their case, k's that yield a high percentage of primes) to maximize their chances of finding one. In your case, it shouldn't be surprising that there are many primes of the form N = 8n^2 - 1. If we write this as 2^3n^2 - 1, we see that unless n = 2^j and 2j+3 is composite (i.e. a composite-exponent Mersenne), the fact that the power on the 2 and the n are relatively prime means we can't write N as a^k - b^k (in your case b = 1), which rules out any easy algebraic factors. That significantly boosts the odds of N being prime. It's also quite likely (I'll leave you to investigate this) that N of this form, if they are composite, must have their factors of a certain algebraic form. That would again increase the prime density, by eliminating most random odd numbers as candidate factors. Yes, the Mersennes are a special case of these numbers, but it seems doubtful that fact will help us find more Mersennes - those are a special case unto themselves, we know a lot about their properties and even have a fast deterministic primality test for them. We also know they occur (AFAWK) for random prime exponents of 2, and that their density thins out fairly predictably as they get bigger. FYI, here are all the n <= 10^4 for which 2^3*n^2 - 1 is prime. You see the general thinning-out trend, with a few hiccups (e.g. the n = 5000-6000 and 8000-9000 columns are longer than the ones that precede them), but one sees such behavior for most any reasonably dense sequence of primes one wishes to investigate: [code:1] 1 1003 2004 3013 4006 5012 6001 7003 8008 9016 2 1004 2012 3014 4008 5014 6002 7007 8012 9018 3 1008 2020 3015 4011 5015 6004 7018 8013 9027 4 1011 2021 3017 4037 5017 6013 7023 8018 9033 5 1018 2023 3022 4046 5033 6017 7037 8029 9040 9 1026 2026 3027 4048 5035 6020 7038 8032 9041 11 1032 2034 3031 4057 5036 6024 7049 8039 9042 12 1033 2035 3035 4058 5052 6027 7051 8040 9056 14 1034 2051 3038 4067 5058 6029 7056 8046 9065 17 1036 2053 3049 4071 5065 6034 7061 8052 9074 18 1038 2068 3057 4074 5068 6037 7063 8060 9077 19 1039 2072 3061 4076 5075 6044 7067 8075 9082 21 1041 2083 3062 4077 5085 6052 7072 8078 9086 23 1045 2090 3068 4081 5092 6055 7094 8081 9089 25 1048 2095 3077 4084 5098 6060 7098 8087 9090 26 1050 2103 3080 4088 5103 6072 7102 8101 9093 28 1053 2104 3096 4091 5106 6080 7105 8104 9097 31 1055 2107 3105 4095 5112 6083 7117 8109 9100 32 1066 2112 3113 4111 5120 6086 7119 8113 9109 38 1069 2114 3115 4113 5121 6087 7124 8122 9110 40 1075 2117 3117 4116 5128 6094 7142 8127 9114 46 1080 2119 3119 4118 5135 6101 7143 8134 9118 49 1082 2124 3125 4123 5143 6107 7144 8137 9121 51 1085 2128 3140 4128 5150 6114 7152 8139 9130 54 1092 2139 3143 4132 5152 6116 7156 8144 9131 56 1094 2140 3145 4137 5154 6118 7157 8145 9132 59 1099 2142 3153 4139 5170 6121 7158 8160 9144 63 1102 2145 3154 4142 5183 6122 7165 8174 9145 66 1104 2147 3157 4144 5184 6129 7170 8176 9146 67 1106 2151 3161 4148 5190 6135 7171 8185 9166 70 1122 2153 3168 4154 5194 6137 7178 8197 9179 77 1125 2154 3175 4160 5197 6141 7179 8202 9182 79 1127 2158 3178 4161 5199 6142 7180 8207 9189 80 1131 2160 3180 4163 5204 6146 7189 8214 9203 82 1136 2165 3187 4177 5220 6150 7199 8229 9210 86 1137 2175 3201 4191 5231 6155 7200 8232 9223 89 1148 2180 3202 4198 5233 6158 7203 8234 9235 93 1150 2191 3209 4200 5240 6160 7212 8242 9242 94 1152 2196 3210 4203 5247 6162 7214 8253 9244 96 1153 2210 3230 4204 5252 6165 7220 8265 9245 98 1158 2214 3232 4207 5255 6170 7224 8267 9250 100 1159 2226 3234 4210 5261 6186 7227 8270 9259 102 1164 2229 3241 4211 5273 6190 7231 8271 9268 103 1169 2236 3255 4228 5275 6193 7234 8274 9278 107 1172 2240 3262 4230 5281 6199 7236 8277 9282 110 1174 2242 3264 4237 5282 6205 7240 8278 9287 114 1178 2243 3266 4247 5285 6211 7241 8281 9291 116 1179 2244 3278 4251 5287 6216 7254 8283 9293 119 1181 2252 3279 4252 5296 6218 7255 8284 9303 121 1187 2256 3283 4266 5302 6220 7259 8292 9305 124 1188 2257 3304 4272 5309 6223 7262 8298 9315 128 1190 2259 3313 4281 5310 6234 7264 8302 9317 133 1192 2264 3314 4282 5313 6235 7270 8313 9319 135 1193 2272 3320 4286 5317 6237 7276 8316 9324 137 1201 2273 3321 4293 5332 6239 7277 8325 9327 140 1213 2277 3323 4295 5341 6265 7291 8328 9328 144 1215 2278 3329 4298 5346 6268 7296 8339 9335 147 1218 2279 3332 4300 5353 6269 7319 8344 9349 150 1221 2280 3341 4301 5355 6275 7324 8346 9362 152 1223 2286 3344 4303 5357 6276 7326 8347 9366 156 1225 2289 3348 4305 5360 6289 7338 8348 9370 161 1227 2291 3358 4309 5371 6293 7340 8355 9378 166 1228 2299 3362 4316 5373 6302 7357 8361 9382 173 1229 2312 3364 4321 5376 6304 7359 8362 9385 182 1239 2315 3367 4326 5386 6307 7360 8384 9389 186 1241 2317 3371 4329 5388 6310 7361 8389 9396 187 1243 2320 3379 4330 5390 6312 7375 8398 9404 189 1250 2321 3384 4333 5401 6319 7378 8407 9405 192 1253 2324 3392 4344 5402 6337 7382 8412 9413 193 1260 2326 3399 4349 5409 6338 7383 8417 9417 201 1263 2333 3414 4350 5411 6339 7392 8418 9439 203 1264 2334 3416 4351 5414 6347 7394 8419 9447 205 1267 2335 3418 4358 5436 6352 7395 8424 9455 207 1271 2338 3428 4370 5437 6370 7396 8426 9464 208 1276 2342 3430 4371 5439 6372 7397 8428 9467 213 1283 2357 3432 4375 5441 6375 7406 8432 9468 220 1288 2359 3437 4378 5443 6379 7408 8433 9471 221 1292 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6498 7516 8545 9586 278 1375 2459 3531 4491 5553 6499 7518 8551 9593 285 1376 2461 3538 4496 5560 6506 7520 8559 9594 290 1377 2464 3542 4499 5561 6507 7522 8565 9597 291 1382 2467 3551 4508 5562 6515 7536 8568 9599 301 1389 2473 3556 4517 5565 6517 7548 8571 9604 303 1414 2483 3565 4531 5568 6522 7556 8577 9606 304 1425 2485 3566 4533 5572 6524 7563 8579 9609 305 1426 2487 3570 4548 5579 6527 7565 8580 9613 310 1430 2494 3574 4552 5597 6536 7569 8582 9618 311 1431 2499 3586 4553 5602 6545 7585 8593 9622 317 1437 2502 3588 4567 5607 6559 7588 8596 9634 318 1439 2504 3589 4571 5614 6573 7598 8599 9635 325 1449 2510 3591 4587 5618 6582 7599 8603 9650 326 1453 2511 3596 4588 5624 6583 7607 8605 9651 327 1460 2515 3615 4589 5625 6584 7621 8608 9657 345 1463 2516 3616 4602 5632 6591 7625 8614 9662 348 1475 2520 3617 4608 5635 6598 7633 8615 9664 353 1477 2524 3622 4613 5638 6604 7644 8628 9665 354 1480 2529 3626 4615 5645 6611 7653 8635 9671 355 1481 2530 3630 4620 5646 6612 7658 8650 9678 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8831 9896 516 1685 2719 3805 4806 5864 6846 7875 8832 9905 521 1698 2723 3808 4809 5866 6848 7877 8844 9908 523 1699 2735 3820 4811 5873 6867 7879 8851 9914 532 1704 2740 3822 4813 5876 6876 7887 8852 9915 541 1706 2746 3826 4819 5884 6885 7899 8853 9924 542 1711 2748 3827 4839 5894 6888 7910 8857 9926 544 1712 2751 3829 4840 5911 6897 7917 8860 9933 548 1713 2754 3833 4842 5922 6898 7924 8866 9942 549 1720 2756 3854 4848 5924 6900 7927 8869 9944 553 1722 2770 3867 4858 5931 6913 7935 8888 9951 555 1731 2793 3868 4861 5955 6921 7940 8897 9964 563 1733 2796 3875 4870 5968 6925 7941 8902 9987 564 1740 2797 3876 4874 5988 6928 7943 8906 9991 567 1743 2802 3881 4875 5990 6941 7945 8907 9992 569 1745 2804 3885 4877 5995 6944 7954 8911 9999 570 1747 2817 3887 4881 5997 6948 7961 8913 10000 572 1752 2818 3892 4882 5999 6949 7962 8923 577 1753 2821 3895 4884 6951 7964 8927 578 1754 2824 3899 4891 6956 7975 8936 579 1759 2825 3906 4905 6968 7978 8941 581 1762 2828 3910 4907 6979 7985 8942 583 1768 2831 3915 4909 6982 7998 8948 584 1773 2840 3916 4917 6984 8956 593 1788 2844 3922 4918 6986 8979 599 1789 2851 3936 4921 6996 8981 600 1790 2860 3943 4924 8993 611 1794 2867 3944 4932 8995 613 1797 2870 3945 4933 8998 616 1803 2873 3946 4947 625 1804 2874 3952 4963 640 1811 2875 3953 4966 641 1816 2879 3957 4981 644 1820 2888 3959 4982 646 1822 2889 3964 4987 663 1825 2893 3974 667 1830 2895 3976 669 1834 2896 3981 672 1837 2902 3987 675 1845 2909 3993 676 1850 2929 3999 686 1852 2944 688 1855 2949 693 1857 2950 698 1859 2959 700 1864 2963 705 1865 2964 709 1867 2973 712 1872 2977 714 1878 2986 716 1887 2989 717 1900 2993 718 1904 2996 725 1922 3000 726 1929 730 1930 735 1935 737 1944 739 1946 745 1956 747 1957 749 1958 751 1970 753 1976 754 1977 760 1978 767 1981 770 1983 774 1990 779 781 787 793 796 802 805 816 822 824 829 833 845 850 851 865 871 872 873 879 882 887 892 898 899 900 905 907 910 914 915 919 922 935 938 947 954 961 964 968 971 975 983 984 985 987 998 [/code:1] Re. your comment about N = 2^k - 1 for non-integer k: if I allow k to be irrational, I can write any positive integer I want this way. So what?

2002-11-06, 02:41	#8
TTn 2²·3·617 Posts	Thanks for the list! I doubt, the doubt. The comment 2^k-1 was to verify that we were not on the same page.