For the Riesel base b problem, a k has algebra factors if and only if k*b^n is a perfect power (of the form m^r with r>1) for some n, and for Sierpinski base b problem, a k has algebra factors if and only if k*b^n is either a perfect odd power (of the form m^r with odd r>1) or of the form 4*m^4 for some n.

[QUOTE=sweety439;467154]For the Riesel base b problem, a k has algebra factors if and only if k*b^n is a perfect power (of the form m^r with r>1) for some n, and for Sierpinski base b problem, a k has algebra factors if and only if k*b^n is either a perfect odd power (of the form m^r with odd r>1) or of the form 4*m^4 for some n.[/QUOTE]

If a k has algebra factors, you should check this k for possiblity for having a prime. (if and only if this k has no possiblity for having a prime, then this k is excluded from the conjecture)

For example:

(343*15^n+1)/2 (has prime candidate), 4*53^n+1 (has prime candidate), (343*63^n+1)/2 (has prime candidate), 8*86^n+1 (has prime candidate), 1024*30^n-1 (has prime candidate), 1369*30^n-1 (proven composite by partial algebra factors), (1600*42^n-1)/41 (has prime candidate), (100*46^n-1)/9 (has prime candidate), 384*48^n-1 (has prime candidate), (169*58^n-1)/3 (has prime candidate), (400*58^n-1)/57 (has prime candidate), (100*61^n-1)/3 (has prime candidate), (25*67^n-1)/6 (has prime candidate), (169*85^n-1)/84 (has prime candidate), (49*88^n-1)/3 (has prime candidate), (400*88^n-1)/3 (proven composite by partial algebra factors), (27*91^n-1)/2 (has prime candidate).

A classic example for the k proven composite by partial algebra factors is (343*10^n-1)/9, but this k is above the CK for this base.

And for the k's having known large primes (smallest exponent > 5K), such as 32*26^n+1, 4*77^n+1, 4*83^n+1, 32*26^n-1, 1024*60^n-1, 729*70^n-1, 4*72^n-1, etc.

Found a prime:

36*93^3936+1

Thus, k=36 can be removed from the remain k's for S93.

S93 now has 6 k's remain: 19, 43, 62, 67, 87, 93 (k=62 has been tested to n=100K by CRUS project, other k's are only at n=1K).

[QUOTE=sweety439;467063]My computer is now searching for generalized repunit primes (b^n-1)/(b-1) (see [URL="http://mersenneforum.org/showthread.php?t=21808"]http://mersenneforum.org/showthread.php?t=21808[/URL]), I will reserve this project (extended SR problems) after these searched were done.[/QUOTE]

I will reserve [I]all[/I] k's for [I]all[/I] Sierpinski/Riesel bases b <= 128 (except bases 66 and 120) and b = 256, 512 and 1024 to n=5K, also reserve [I]all[/I] Sierpinski/Riesel bases with <= 10 k's remaining at n=5K to n=25K, especially S10, S25, S33, R31 and R33.

Reserve S103 and R97 (only for the k's not in CRUS).

For S97 k=22, I found the prime 22*97^2182+1, S97 k=64 is still reserving...

[QUOTE=sweety439;468289]Reserve S103 and R97 (only for the k's not in CRUS).[/QUOTE]

(16*97^1627-1)/3 is (probable) prime!!!

S106 has totally 42 k's remain at n=1K:

69, 110, 164, 198, 259, 412, 436, 449, 635, 653, 679, 740, 748, 812, 887, 929, 1000, 1088, 1160, 1190, 1421, 1429, 1511, 1544, 1559, 1607, 1628, 1703, 1796, 1823, 1835, 1925, 1973, 1985, 2018, 2036, 2075, 2119, 2177, 2189, 2216, 2279

R127 has totally 224 k's remain at n=1K:

13, 17, 25, 27, 33, 35, 79, 83, 91, 113, 121, 139, 159, 179, 191, 231, 233, 235, 236, 237, 239, 250, 251, 264, 279, 288, 293, 333, 353, 361, 367, 379, 443, 451, 459, 471, 473, 511, 513, 517, 523, 531, 537, 551, 553, 557, 561, 597, 599, 604, 617, 631, 639, 649, 659, 679, 699, 715, 725, 731, 733, 737, 739, 747, 751, 755, 763, 773, 778, 783, 797, 809, 838, 848, 863, 871, 895, 919, 937, 939, 950, 953, 964, 982, 997, 999, 1013, 1019, 1025, 1031, 1037, 1039, 1043, 1051, 1106, 1107, 1117, 1119, 1127, 1157, 1173, 1185, 1196, 1199, 1211, 1231, 1232, 1233, 1245, 1253, 1259, 1279, 1288, 1291, 1313, 1327, 1333, 1335, 1337, 1347, 1353, 1359, 1371, 1377, 1401, 1407, 1417, 1421, 1429, 1432, 1439, 1473, 1481, 1491, 1513, 1525, 1539, 1549, 1551, 1573, 1577, 1579, 1589, 1593, 1595, 1597, 1599, 1611, 1612, 1618, 1631, 1639, 1641, 1651, 1661, 1677, 1693, 1699, 1709, 1711, 1731, 1732, 1737, 1751, 1771, 1792, 1793, 1803, 1837, 1839, 1903, 1911, 1921, 1928, 1933, 1936, 1939, 1943, 1951, 1957, 1959, 1999, 2013, 2017, 2032, 2039, 2045, 2072, 2073, 2079, 2092, 2097, 2099, 2129, 2155, 2159, 2168, 2179, 2191, 2197, 2215, 2231, 2247, 2253, 2273, 2279, 2303, 2313, 2339, 2367, 2377, 2389, 2411, 2427, 2431, 2433, 2479, 2501, 2543, 2548, 2559, 2565, 2573, 2583

[QUOTE=sweety439;468297]R127 has totally 224 k's remain at n=1K:

13, 17, 25, 27, 33, 35, 79, 83, 91, 113, 121, 139, 159, 179, 191, 231, 233, 235, 236, 237, 239, 250, 251, 264, 279, 288, 293, 333, 353, 361, 367, 379, 443, 451, 459, 471, 473, 511, 513, 517, 523, 531, 537, 551, 553, 557, 561, 597, 599, 604, 617, 631, 639, 649, 659, 679, 699, 715, 725, 731, 733, 737, 739, 747, 751, 755, 763, 773, 778, 783, 797, 809, 838, 848, 863, 871, 895, 919, 937, 939, 950, 953, 964, 982, 997, 999, 1013, 1019, 1025, 1031, 1037, 1039, 1043, 1051, 1106, 1107, 1117, 1119, 1127, 1157, 1173, 1185, 1196, 1199, 1211, 1231, 1232, 1233, 1245, 1253, 1259, 1279, 1288, 1291, 1313, 1327, 1333, 1335, 1337, 1347, 1353, 1359, 1371, 1377, 1401, 1407, 1417, 1421, 1429, 1432, 1439, 1473, 1481, 1491, 1513, 1525, 1539, 1549, 1551, 1573, 1577, 1579, 1589, 1593, 1595, 1597, 1599, 1611, 1612, 1618, 1631, 1639, 1641, 1651, 1661, 1677, 1693, 1699, 1709, 1711, 1731, 1732, 1737, 1751, 1771, 1792, 1793, 1803, 1837, 1839, 1903, 1911, 1921, 1928, 1933, 1936, 1939, 1943, 1951, 1957, 1959, 1999, 2013, 2017, 2032, 2039, 2045, 2072, 2073, 2079, 2092, 2097, 2099, 2129, 2155, 2159, 2168, 2179, 2191, 2197, 2215, 2231, 2247, 2253, 2273, 2279, 2303, 2313, 2339, 2367, 2377, 2389, 2411, 2427, 2431, 2433, 2479, 2501, 2543, 2548, 2559, 2565, 2573, 2583[/QUOTE]

The k's in the same family (then they will have the same prime) are:
{13, 1651}
{17, 2159}

Thus, R127 has in fact only 222 k's remain at n=1K:

13, 17, 25, 27, 33, 35, 79, 83, 91, 113, 121, 139, 159, 179, 191, 231, 233, 235, 236, 237, 239, 250, 251, 264, 279, 288, 293, 333, 353, 361, 367, 379, 443, 451, 459, 471, 473, 511, 513, 517, 523, 531, 537, 551, 553, 557, 561, 597, 599, 604, 617, 631, 639, 649, 659, 679, 699, 715, 725, 731, 733, 737, 739, 747, 751, 755, 763, 773, 778, 783, 797, 809, 838, 848, 863, 871, 895, 919, 937, 939, 950, 953, 964, 982, 997, 999, 1013, 1019, 1025, 1031, 1037, 1039, 1043, 1051, 1106, 1107, 1117, 1119, 1127, 1157, 1173, 1185, 1196, 1199, 1211, 1231, 1232, 1233, 1245, 1253, 1259, 1279, 1288, 1291, 1313, 1327, 1333, 1335, 1337, 1347, 1353, 1359, 1371, 1377, 1401, 1407, 1417, 1421, 1429, 1432, 1439, 1473, 1481, 1491, 1513, 1525, 1539, 1549, 1551, 1573, 1577, 1579, 1589, 1593, 1595, 1597, 1599, 1611, 1612, 1618, 1631, 1639, 1641, 1661, 1677, 1693, 1699, 1709, 1711, 1731, 1732, 1737, 1751, 1771, 1792, 1793, 1803, 1837, 1839, 1903, 1911, 1921, 1928, 1933, 1936, 1939, 1943, 1951, 1957, 1959, 1999, 2013, 2017, 2032, 2039, 2045, 2072, 2073, 2079, 2092, 2097, 2099, 2129, 2155, 2168, 2179, 2191, 2197, 2215, 2231, 2247, 2253, 2273, 2279, 2303, 2313, 2339, 2367, 2377, 2389, 2411, 2427, 2431, 2433, 2479, 2501, 2543, 2548, 2559, 2565, 2573, 2583

S73, the last k tested to n=10K, no (probable) prime found.