Loeschian Primes

I wrote about Loeschian numbers in an eponymous post on January 5th 2022 where I wrote that Loeschian numbers are numbers of the form \(x^2+xy+y^2\) where \(x\) and \(y\) are positive integers. These integers do not need to be distinct and do not need to be prime. Of the first one thousand integers, 277 of them are Loeschian and so they are relatively common.

However, if we restrict \(x\), \(y\) and the Loeschian number itself to being prime then the resultant Loeschian primes form OEIS A244146:

A244146

Primes of the form \(x^2 + x \times y + y^2\) with \(x\), \(y\) primes.

The initial members of the sequence are (permalink):

19, 67, 79, 109, 163, 199, 349, 433, 457, 607, 691, 739, 937, 997, 1063, 1093, 1327, 1423, 1447, 1489, 1579, 1753, 1777, 1987, 2017, 2089, 2203, 2287, 2383, 2749, 3229, 3463, 3847, 3943, 4051, 4177, 4513, 4567, 5347, 5413, 5479, 5557, 5653, 6079, 6133, 6271, 6661, 7537, 7867, 7873, 8287, 9973, 10513, 10597, 10957, 11149, 11161, 11329, 11779, 11863, 11923, 12007, 12163, 12451, 12583, 13009, 13063, 13093, 13597, 14293, 14407, 14437, 15277, 15307, 15727, 16519, 17359, 17659, 18127, 19213, 19477, 19603, 19687, 19777, 20011, 20719, 21193, 22963, 23017, 23293, 23581, 23977, 24421, 24967, 25657, 26347, 26497, 26701, 27067, 27103, 27409, 27697, 27817, 28789, 29629, 30187, 30427, 31189, 31237, 32587, 33091, 33343, 33577, 33613, 33751, 34231, 34537, 35353, 35437, 35803, 36919, 37699, 37747, 38239, 38851, 39133, 39157, 39217, 39409, 39667, 40213, 40237, 40471, 40819, 41263, 41467, 41539, 41863, 41911, 42223, 43321, 44203, 44917, 46681, 47797, 48157, 48523, 49927, 50287, 50707, 51853, 52957, 53233, 54547, 54673, 57973, 58771, 59743, 62233, 62347, 63577, 64927, 65983, 66763, 66943, 69709, 70423, 74857, 76243, 78853, 79693, 81931, 82351, 84697, 84793, 85513, 91453, 97369, 103813, 103837, 107227

It can be noted that \(x\) and \(y\) cannot be equal otherwise we have \(3x^2\). The first Loeschian prime is 19 and is constructed as follows:$$2^2+2 \times 3 + 3^2=19$$The last prime in the above list is 107227 and is constructed as follows:$$181^2+181 \times 197 + 197^2= 107227$$Interestingly none of these primes are Sophie Germain primes. If we extend the range to one million there are still none. 

However, if we look for primes of the form \(2p-1\) rather than \(2p+1\), we find many. In the range up to about a quarter of a million, there are 1706 Loeschian primes and, of these, 309 are of the form \(2p-1\). These primes are:

19, 79, 199, 607, 691, 937, 997, 2089, 4051, 4177, 4567, 5479, 5557, 6079, 6271, 7537, 7867, 8287, 10597, 11779, 13009, 14407, 15277, 16519, 17659, 19477, 19687, 23581, 26701, 27067, 27817, 30187, 31237, 32587, 33577, 34537, 39667, 40237, 40819, 41539, 44917, 52957, 57037, 57487, 57847, 58567, 58771, 62347, 64927, 69247, 74527, 74857, 75781, 77569, 81931, 82351, 87697, 89071, 89767, 92347, 94219, 95317, 96757, 97327, 98947, 102241, 103837, 108517, 109717, 113329, 114859, 114889, 116191, 127657, 129967, 133261, 133717, 135799, 137791, 154579, 160861, 161527, 164419, 164617, 170197, 172801, 174877, 174907, 181081, 188827, 190837, 201997, 210229, 215767, 216481, 218749, 226267, 226777, 228577, 229267, 232861, 233917, 234457, 238639, 239137, 244159, 244567, 245269, 247519, 248407, 255181, 256579, 259717, 261631, 264991, 280327, 293767, 298159, 303151, 310567, 311407, 311827, 315097, 321187, 325477, 328921, 332617, 336901, 343087, 351457, 351517, 362407, 364537, 367687, 377617, 378997, 382549, 385027, 385057, 402037, 406117, 406789, 407857, 409177, 416947, 423769, 428167, 430747, 435307, 436957, 443089, 454849, 455647, 457981, 458317, 464467, 467017, 470461, 473287, 486769, 487717, 500911, 519301, 525127, 531457, 534241, 534637, 541267, 542149, 544477, 547237, 555967, 564667, 568807, 575557, 580747, 610447, 620947, 621739, 630967, 633037, 634597, 640837, 644257, 651727, 652417, 655399, 673951, 693097, 703897, 705787, 706837, 709117, 710557, 739507, 741847, 761977, 763897, 769597, 771769, 778237, 780817, 784129, 796267, 803911, 811777, 813097, 813217, 823399, 826759, 830719, 832747, 841207, 857029, 858427, 867487, 877567, 888397, 888541, 894547, 897601, 903757, 908671, 909289, 912337, 916417, 918079, 924139, 956107, 960217, 970867, 971077, 974137, 975847, 986149, 987061, 989887, 992449, 999067, 1009669, 1018447, 1024477, 1024987, 1027459, 1030867, 1037857, 1044457, 1050391, 1050811, 1058221, 1059547, 1061317, 1112341, 1113667, 1130359, 1132309, 1137457, 1139227, 1145269, 1155997, 1192267, 1203667, 1209337, 1231177, 1248271, 1257787, 1260577, 1283677, 1285777, 1294597, 1332547, 1335361, 1351117, 1385947, 1395127, 1442377, 1457647, 1468507, 1472077, 1485937, 1522447, 1559689, 1563817, 1572871, 1609477, 1620739, 1631491, 1635559, 1662457, 1688497, 1693429, 1699381, 1712149, 1724617, 1755739, 1795867, 1831441, 1900687, 1908367, 1911037, 1959697, 2028277, 2103307, 2116969, 2193337, 2232427, 2407117, 2413927, 2599189

Take the first as an example:$$19 \times 2 -1=37 \text{ which is prime}$$If we look at the final number in the list above, we see that:$$25599189 \times 2 -1=51198377 \text{ which is prime}$$Of all these Loeschian and \(2p-1\) primes above, only one leads to a prime that is also Loeschian and that prime is 2089 leading to 4177. Here are their constructions:$$ \begin{align} 2089&=5^2 + 5 \times 43 + 43^2\\4177 &= 2089 \times 2 -1\\&=19^2+19 \times 53 +53^2 \end{align}$$We have to go to 3296949 before this happens again:$$ \begin{align} 3291949 &=397^2+ 397 \times 1583+1583^2\\ 6583897 &=3291949 \times 2 -1\\ &=53^2+53 \times 2539 + 2539^2 \end{align}$$If we extend our range still further we find the following: 27793477, 65947201, 87196177, 88718437, 160502137 and 172502377. Clearly such numbers are few and far between (permalink).