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I've been using this equation to find primes of a specified range, in this example the range is 10 to 10000. Go ahead and try it out, your monitor will not catch fire. The problem I'm now having is I'm getting false positives when the computer searches for 0002 and it comes across a 0022, it will give a false positive that is a fraction. I've included the final output so you can see the fractions. It will be a high number over the spacing, a 4 in this example.

N[Sum[2/10^(Mod[-(10 - n^2), n]*4)/(10^(n*4) - 1), {n, 1, 10000^(1/2)}], (10000 - 10)*4]
str = StringDrop[ToString[%], 2]
Last /@ StringPosition[str, "0002"]/4 + 10

{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, \
79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, \
151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, \
229, 233, 239, 241, 251, 257, 263, 269, 271, 1103/4, 277, 1119/4, \
281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, \
373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, \
457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, \
557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 2463/4, 617, \
619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, \
719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, \
821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, \
911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, \
3999/4, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, \
1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, \
1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, \
1229, 1231, 1237, 4959/4, 1249, 1259, 1277, 1279, 1283, 1289, 1291, \
1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, \
1399, 5631/4, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, \
5823/4, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 5983/4, 1499, 1511, \
1523, 1531, 1543, 6191/4, 1549, 1553, 1559, 1567, 1571, 1579, 1583, \
1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, \
1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, \
6999/4, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, \
1847, 1861, 1867, 7479/4, 1871, 1873, 1877, 1879, 1889, 1901, 1907, \
1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, \
2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, \
2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, \
2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 9023/4, \
2267, 2269, 2273, 2281, 2287, 2293, 9183/4, 2297, 2309, 2311, 2333, \
2339, 2341, 9383/4, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, \
2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, \
2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 10223/4, 2557, 2579, \
2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, \
2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, \
2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, \
2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, \
2909, 11663/4, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, \
3001, 12011/4, 12039/4, 3011, 3019, 3023, 3037, 3041, 3049, 3061, \
3067, 3079, 3083, 3089, 3109, 3119, 3121, 12543/4, 3137, 3163, 3167, \
3169, 3181, 3187, 3191, 3203, 3209, 3217, 12879/4, 3221, 3229, 3251, \
3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, \
3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, \
3433, 3449, 3457, 3461, 3463, 3467, 3469, 13887/4, 3491, 3499, 3511, \
3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, \
3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, \
3673, 3677, 3691, 3697, 3701, 3709, 3719, 14895/4, 3727, 3733, 3739, \
3761, 3767, 3769, 3779, 15135/4, 3793, 3797, 3803, 3821, 3823, \
15299/4, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 15623/4, \
3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, \
15983/4, 15999/4, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 16191/4, \
4049, 4051, 4057, 4073, 4079, 4091, 4093, 16379/4, 4099, 4111, 4127, \
4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, \
4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 17111/4, 4283, \
4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, \
4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, \
4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, \
4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, \
4679, 4691, 18791/4, 4703, 4721, 4723, 4729, 18927/4, 4733, 4751, \
4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 19295/4, 4831, \
4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, \
4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, \
5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, \
5113, 5119, 20543/4, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, \
5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, \
5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, \
5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 21951/4, \
5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, \
5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, \
5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, \
5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, \
5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, \
5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, \
6073, 24303/4, 6079, 24335/4, 6089, 6091, 6101, 6113, 6121, 6131, \
6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, \
6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, \
6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, \
6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, \
6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, \
6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, \
6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, \
6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, \
6907, 6911, 6917, 27743/4, 6947, 6949, 6959, 6961, 6967, 6971, 6977, \
6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, \
7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, \
7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, \
7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, \
7433, 7451, 7457, 7459, 29879/4, 7477, 7481, 7487, 7489, 7499, 7507, \
7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, \
7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, \
7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 30975/4, 7753, \
7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, \
7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, \
7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, \
8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, \
8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, \
8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, \
8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, \
8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, \
8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, \
8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, \
8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, \
8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, \
8971, 8999, 9001, 9007, 36035/4, 9011, 9013, 9029, 9041, 9043, 9049, \
9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, \
9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, \
9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, \
9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, \
9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, \
9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, \
9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, \
9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, \
9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, \
9931, 9941, 9949, 9967, 9973}
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  • $\begingroup$ Even eliminating the "00022" mismatches still results in some non-integer values getting spit out, corresponding to parts of the string where matches are valid by your criteria, e.g., "...040010002000020004001..." in the string. I deleted my answer after observing that, so please clarify precisely the restrictions on matches. $\endgroup$
    – rasher
    Feb 15 '15 at 8:57
  • 3
    $\begingroup$ I am going to vote to close this, but want to explain why. Your interest in finding large primes is admirable, and even amateurs have made breakthroughs, c.f. Eric Rowland. However, at this point it would be better to spend some time on Math.StackExchange to validate (or invalidate) the tack you're on. It is easy to get the full expansion of your sum and show that applying your formula on it fails at this time. At the same time, perhaps read the Mathematica documentation, guides, tutorials, and the guides at this very site. The basic operations you've queried about are all covered within. $\endgroup$
    – rasher
    Feb 15 '15 at 10:16

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