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Old 2020-09-18, 20:38   #982
sweety439
 
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Nov 2016

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k = 676:

since 676 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 17 · 281
3 : 3 · 277 · 64591
5 : 5 · 17 · 19 · 67 · 5573621
7 : 1949 · 3476839593221
9 : 3^3 · 17 · 3271 · 50712496951637
11 : 19 · 42937 · 2432147 · 431166327217
13 : 17 · 1373 · 6351547249 · 64838460350149
17 : 17 · 19 · 61 · 61591784776543827671882518345783
19 : 6299 · 756585273193 · 2861128642099661938794059
23 : 19 · 98443 · 920347627017000007051307391604325416676033
43 : (a 89-digit composite with no known prime factor)
67 : 2843 · (a 134-digit composite with no known prime factor)
79 : 1129 · 32491 · (a 155-digit composite with no known prime factor)
91 : 105899 · (a 181-digit composite with no known prime factor)

Although this number is divisible by 3 for all n == 3 mod 6 and by 19 for all n == 5 mod 6 and by 17 for all n == 1 mod 4 (which makes this k-value very low weight, since only n == 7 mod 12 can be prime), but it does not appear to be any covering set of primes, so there must be a prime at some point.

k = 841:

since 841 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 3 · 283
3 : 271 · 35201
5 : 61 · 677 · 2595479
7 : 3^2 · 5^2 · 5352605493383
9 : 4679 · 8663 · 333839809991
11 : 67 · 11440889 · 198352025576693
15 : 359487408541 · 53396278847280064403
17 : 5 · 19927 · 140909 · 15362282538731494849528849
21 : 271 · 7457 · 663563 · 20305527277370848392217057350779
23 : 94547 · 534824108672537 · 6050383020924045192372407269
29 : 84737 · (a 55-digit prime)
33 : 311 · 1888306597 · 1129552782935923 · 2923571188269551 · 28251866661502752658291361
51 : 2843 · (a 101-digit composite with no known prime factor)
53 : 13456811 · 88286677 · 6437291630956799 · (a 78-digit prime)
59 : (a 121-digit composite with no known prime factor)
63 : 2371 · 6059059478263861 · (a 110-digit composite with no known prime factor)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

k = 1024:

since 1024 is square and 5-th power, all even n and all n divisible by 5 have algebra factors, and we only want to know whether it has a covering set of primes for all n == 1, 3, 7, 9 (mod 10), if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 97 · 373
3 : 17 · 3407 · 7019
7 : 17 · 3019053696484613
9 : 647 · 3581827 · 248841380929
11 : 3 · 17^2 · 7473501436891484179943
13 : 449 · 1447 · 112057280449127255045987
17 : 3^2 · 406591 · 2126171 · 1181353712721405831409129
19 : 17 · 283 · 258373 · 9179867 · 9050472811960369021895401
21 : 1831 · 972605267 · 1597539586927967 · 407873305308400559
33 : 1223 · (a 67-digit prime)
37 : 7753 · 2460302303 · (a 65-digit prime)
49 : 97 · 839 · 25561 · 136811 · 45385621130173559982180883 · (a 62-digit prime)
57 : 1777 · 66191 · 10482163 · 4863222893 · (a 94-digit prime)
61 : (a 127-digit composite with no known prime factor)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

Last fiddled with by sweety439 on 2020-09-20 at 21:58
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