## Friday, October 07, 2016

### [abymulnr] Primes in the neighborhood of powers of two

Prime numbers in the vicinity of "nice" powers of two.  The exponent is itself of the form 2^n, 3*2^n, or 5*2^n.  The larger primes were discovered with Pari/GP.  The gap between 2^65536-5627 and 2^65536-99267 took about 2.8 days.  2^65536 = 2^2^2^2^2.

Primes were verified additionally with 20 iterations of Miller-Rabin, with randomly chosen bases.

Search ranges: 2^2^15 -149000 +227000, 2^2^16 -154000 +163000.  For others, the search stopped at the last number.

Related OEIS: A058220 A058221 A129786.

Previously: Smaller numbers, Twin primes

2^1 +0 +1 +3 +5 +9 +11 +15 +17 +21 +27

2^2 -1 -2

2^2 +1 +3 +7 +9 +13 +15 +19 +25 +27 +33

2^3 -1 -3 -5 -6

2^3 +3 +5 +9 +11 +15 +21 +23 +29 +33 +35

2^4 -3 -5 -9 -11 -13 -14

2^4 +1 +3 +7 +13 +15 +21 +25 +27 +31 +37

2^5 -1 -3 -9 -13 -15 -19 -21 -25 -27 -29

2^5 +5 +9 +11 +15 +21 +27 +29 +35 +39 +41

2^6 -3 -5 -11 -17 -21 -23 -27 -33 -35 -41

2^6 +3 +7 +9 +15 +19 +25 +33 +37 +39 +43

2^8 -5 -15 -17 -23 -27 -29 -33 -45 -57 -59

2^8 +1 +7 +13 +15 +21 +25 +27 +37 +51 +55

2^10 -3 -5 -11 -15 -27 -33 -41 -47 -53 -57

2^10 +7 +9 +15 +25 +27 +37 +39 +45 +63 +67

2^12 -3 -5 -17 -23 -39 -45 -47 -69 -75 -77

2^12 +3 +15 +31 +33 +37 +43 +57 +61 +63 +81

2^16 -15 -17 -39 -57 -87 -89 -99 -113 -117 -123

2^16 +1 +3 +7 +15 +21 +27 +43 +45 +51 +63

2^20 -3 -5 -17 -27 -59 -69 -129 -143 -153 -185

2^20 +7 +13 +25 +33 +37 +51 +57 +85 +105 +127

2^24 -3 -17 -33 -63 -75 -77 -89 -95 -117 -167

2^24 +43 +73 +75 +115 +117 +121 +165 +205 +225 +231

2^32 -5 -17 -65 -99 -107 -135 -153 -185 -209 -267

2^32 +15 +61 +75 +81 +91 +93 +163 +181 +201 +217

2^40 -87 -167 -195 -203 -213 -285 -293 -299 -389 -437

2^40 +15 +27 +55 +97 +115 +141 +157 +177 +253 +277

2^48 -59 -65 -89 -93 -147 -165 -189 -233 -243 -257

2^48 +21 +61 +75 +91 +235 +241 +243 +253 +297 +315

2^64 -59 -83 -95 -179 -189 -257 -279 -323 -353 -363

2^64 +13 +37 +51 +81 +93 +141 +307 +331 +393 +493

2^80 -65 -93 -117 -143 -285 -317 -549 -645 -765 -933

2^80 +13 +85 +235 +253 +343 +435 +457 +555 +597 +753

2^96 -17 -87 -93 -147 -165 -189 -237 -243 -315 -347

2^96 +61 +81 +121 +151 +253 +403 +423 +627 +633 +765

2^128 -159 -173 -233 -237 -275 -357 -675 -713 -797 -1193

2^128 +51 +81 +165 +273 +385 +421 +463 +573 +625 +757

2^160 -47 -57 -75 -189 -285 -383 -465 -543 -659 -843

2^160 +7 +291 +357 +421 +471 +501 +643 +685 +861 +921

2^192 -237 -333 -399 -489 -527 -663 -915 -945 -1059 -1143

2^192 +133 +453 +511 +565 +813 +1005 +1045 +1113 +1131 +1423

2^256 -189 -357 -435 -587 -617 -923 -1053 -1299 -1539 -1883

2^256 +297 +301 +357 +487 +583 +757 +795 +807 +847 +931

2^320 -197 -743 -825 -843 -873 -1007 -1017 -1217 -1815 -2955

2^320 +27 +261 +391 +525 +561 +793 +931 +1413 +1857 +1981

2^384 -317 -1437 -1557 -1617 -2147 -2319 -2729 -3087 -3093 -3273

2^384 +231 +331 +417 +535 +735 +817 +823 +835 +933 +1015

2^512 -569 -629 -875 -975 -1695 -1827 -2529 -2807 -2967 -3143

2^512 +75 +145 +285 +727 +1105 +1147 +1273 +2743 +3177 +3913

2^640 -305 -503 -735 -995 -1019 -1215 -1593 -2015 -2033 -2805

2^640 +115 +303 +391 +757 +1287 +1485 +2943 +3627 +3711 +3861

2^768 -825 -1385 -1815 -1845 -2069 -2277 -2825 -3209 -6953 -9189

2^768 +183 +241 +427 +955 +1423 +1723 +1831 +2161 +2851 +3225

2^1024 -105 -179 -1397 -3177 -5025 -5409 -6083 -6369 -6615 -7137

2^1024 +643 +1081 +2113 +2715 +3711 +5335 +5793 +5947 +7015 +7447

2^1280 -1175 -1665 -3149 -5907 -7079 -7607 -7703 -8865 -10043 -10277

2^1280 +1815 +2251 +2553 +3217 +4075 +4405 +4435 +7081 +7405 +9087

2^1536 -3453 -4977 -7769 -8333 -9923 -10859 -11429 -11795 -11843 -11877

2^1536 +75 +1381 +1605 +2487 +4903 +6333 +6637 +7191 +7597 +7681

2^2048 -1557 -2543 -7437 -8507 -9443 -9509 -11339 -11837 -12459 -12855

2^2048 +981 +1617 +3063 +3211 +4143 +7405 +9843 +10665 +10725 +11097

2^2560 -75 -1347 -2745 -6599 -7245 -10205 -12659 -17417 -18413 -20159

2^2560 +903 +4077 +4125 +4363 +5025 +10593 +12153 +13581 +13891 +14767

2^3072 -47 -2165 -4737 -6117 -6489 -7625 -9617 -15269 -23993 -24279

2^3072 +813 +877 +3427 +5131 +9133 +12367 +12433 +13435 +16003 +17245

2^4096 -2549 -8067 -8627 -8799 -9443 -14477 -16859 -17555 -18365 -20655

2^4096 +1761 +7227 +7423 +10093 +10473 +13965 +17335 +17355 +19891 +22803

2^5120 -7097 -7779 -11619 -12689 -15435 -20799 -26745 -28727 -30527 -31955

2^5120 +5467 +6741 +20847 +25135 +26611 +29335 +30147 +33237 +33567 +40791

2^6144 -5157 -6369 -6639 -13773 -13785 -22437 -22923 -27029 -39879 -41895

2^6144 +375 +1275 +2853 +3901 +5545 +11877 +11953 +13171 +14875 +15481

2^8192 -2439 -5619 -9345 -9515 -19085 -19733 -21713 -45933 -46643 -51453

2^8192 +897 +9543 +10813 +13371 +14931 +14985 +15505 +15763 +16305 +19423

2^10240 -323 -10119 -27353 -34179 -40253 -44975 -47195 -66917 -76307 -82187

2^10240 +2601 +15295 +25011 +35671 +46123 +53697 +54387 +58843 +65215 +67963

2^12288 -27803 -30519 -39213 -41045 -43197 -60465 -65055 -72195 -73853 -78515

2^12288 +11293 +17437 +25951 +40413 +40803 +41965 +42775 +46555 +56623 +66175

2^16384 -13797 -29027 -62747 -99125 -101543 -107609 -115673 -118119 -134357 -134705

2^16384 +2775 +36333 +42027 +47025 +55933 +80391 +91395 +95665 +98581 +104565

2^20480 -479 -9159 -21203 -32885 -86435 -88767 -100073 -102509 -104097 -124599

2^20480 +8367 +24201 +28173 +32787 +36061 +46051 +52323 +70725 +72163 +81187

2^24576 -1875 -32507 -37217 -61373 -66905 -67475 -97023 -98079 -103815 -119235

2^24576 +241 +22461 +35193 +80967 +130677 +130965 +145611

2^32768 -25353 -38783 -53373 -61725 -67553 -79613 -112985

2^32768 +118113 +143905 +145027 +160771 +162873 +182005 +187861

2^40960 -48069 -70803

2^40960 +59815 +74031

2^49152 -34689 -37923

2^49152 +1605 +11923

2^65536 -5627 -99267 -123563

2^65536 +44061 +44181 +58227 +106417 +116193 +119031