The Search For First Proth Prime (SNoB Problem)

### Why SNoB?

In April 2020, "JeppeSN" suggested on PrimeGrid forum to extend Sierpiński problem to true Proth primes.

Although we often call any number in form k*b^n+1 a Proth number, strict definition of Proth numbers requires 2^n > k. For classic Sierpiński problem, it does not matter. For example, prime number 12743*2^9+1 is OK to eliminate k=12743 from any of Sierpiński conjectures. But this is not a Proth prime because condition 2^n > k is not met (512 < 12743).

After initial removal of small (up to n=100K) and known primes, we had... exactly 17 k’s left to test! The same number as in original Sierpiński "Seventeen-or-Bust" ("SoB") project. Eventually we started to call our project as "Seventeen New or Bust", or, in short form, "A SNoB project". Probably it was a joke first, but name was accepted by community and became official project name.

The project has three phases, two of them corresponds to existing Sierpiński projects. Their state and findings are posted below. All found primes also were tested for Fermat divisors (currently no divisors were found).

### Pure SNoB

This phase corresponds to original Seventeen-Or-Bust project, testing all k < 78557. Having 17 candidates at start, 10 candidates were quickly eliminated by manual testing.

KPrimeDigitsTested depth (N)
2224922249*2^408602+1123006Up to prime
Found by gd_barnes in 2010, see Note 1
2387323873*2^136733+141166Up to prime
2883128831*2^204580+161590Up to prime
3546135461*2^129820+139085Up to prime
3952739527*2^143055+143069Up to prime
4424344243*2^440969+1132750Up to prime
5495354953*2^622065+1187265Up to prime
5737757377*2^447439+1134698Up to prime
6822168221*2^200944+160496Up to prime
7729777297*2^118499+135677Up to prime

Note 1: this prime was reported by gd_barnes in September 2010, but he used an alternative notation 88996*2^408600+1, so the prime was left unnoticed and was discovered again in this project.

7 remaning candidates were tested on this server using LLR2, which led to discovery of two new primes. The search was stopped at 10M because tasks length started to grow significantly. Interesting that there is a big gap - last prime was found near 2.7M, and there are no primes between 2.7M and 10M.

KPrimeDigitsTested depth (N)
23971-10M
45323-10M
50777-10M
50873-10M
7165771657*2^1146175+1345038Up to prime
76877-10M

### Extra SNoB

Extra SNoB (or SNoB-Extended) is similar to Extended Sierpinski Problem. It includes all K between first and second Sierpinski numbers (from 78557 to 271129). After removal of known T5K primes and quick testing up to 100K, 19 sequences left. They were tested manually using LLR2 up to N = 1M, removing 13 sequences.

KPrimeDigitsTested depth (N)
9022790227*2^138543+141711Up to prime
9715997159*2^523526+1157603Up to prime
130819130819*2^114806+134566Up to prime
145459145459*2^272314+181980420238
(second prime, Note 2)
160817160817*2^756599+1227765Up to prime
165049165049*2^111914+133695Up to prime
165541165541*2^627460+1188890Up to prime
171499171499*2^200746+160436Up to prime
179147179147*2^132227+139810Up to prime
192023192023*2^507229+1152697Up to prime
201031201031*2^170260+151259Up to prime
221989221989*2^351586+1105844Up to prime
248131248131*2^204924+161694Up to prime

Note 2: These tests were run by few people in parallel using same K but different ranges, so two primes were found for K = 145459; second one is 145459*2^420238+1.

No further testing took place yet for 6 remaining sequences:

KPrimeDigitsTested depth (N)
83599-2M
96407-1.5M
97667-2M
129769-2M
149693-2M
225803-1.5M

### MEGA SNoB

MEGA SNoB project includes all K up to 1M. Since definition of Proth prime is 2^n > k, it makes sense to test everything not exactly up to 1000000, but up to 220 = 1048576.

There were 290 sequences after removal of known T5K primes and quick testing up to 100K. Manual testing up to 500K removed 75 sequences (215 remains). Below is a list of found primes (testing depth for these k was up to prime).

KPrimeDigits KPrimeDigits KPrimeDigits
301607301607*2^229647+169137517651517651*2^204528+161575828287828287*2^483751+1145630
308423308423*2^395337+1119014525173525173*2^159553+148036836543836543*2^290465+187445
319531319531*2^212252+163900548869548869*2^304442+191652845899845899*2^338386+1101871
331247331247*2^374775+1112825554573554573*2^305373+191933846857846857*2^453343+1136476
333227333227*2^214471+164568558991558991*2^188204+156661868523868523*2^109737+133041
346223346223*2^373085+1112316575539575539*2^431950+1130036872177872177*2^214575+164600
351199351199*2^149618+145046584971584971*2^266656+180278882077882077*2^478755+1144126
353477353477*2^237219+171416588083588083*2^244477+173601888499888499*2^460922+1138758
357017357017*2^332367+1100058590329590329*2^155334+146766892249892249*2^141518+142608
371791371791*2^144840+143607656063656063*2^133969+140335907043907043*2^305293+191909
374149374149*2^256202+177131693467693467*2^112027+133730911123911123*2^479981+1144495
375121375121*2^132268+139823704507704507*2^321379+196751924683924683*2^421877+1127004
383717383717*2^325171+197892710153710153*2^342801+1103200944011944011*2^372216+1112055
393497393497*2^255283+176854714229714229*2^111318+133516960301960301*2^430616+1129635
396203396203*2^480729+1144720722207722207*2^111831+13367110034411003441*2^191044+157516
397309397309*2^296070+189132722419722419*2^150414+14528510131831013183*2^136521+141103
399617399617*2^340955+1102644752303752303*2^146405+14407910157271015727*2^193231+158175
412501412501*2^279708+184207759653759653*2^154085+14639110246091024609*2^123754+137260
416659416659*2^412866+1124291759947759947*2^108487+13266410285131028513*2^164921+149653
429931429931*2^382776+1115233762769762769*2^188586+15677610342811034281*2^180464+154332
459263459263*2^137237+141319768773768773*2^302509+19107110345991034599*2^233258+170224
467417467417*2^130927+139419773447773447*2^115035+13463510377531037753*2^365805+1110125
479657479657*2^296367+189222774977774977*2^261235+17864610421931042193*2^164893+149644
488341488341*2^466940+1140569802231802231*2^101528+13056910432371043237*2^371971+1111981
495979495979*2^480286+1144587812117812117*2^141051+14246710476611047661*2^382784+1115236

Remaining 215 sequences were tested up to 1M. This work was finished in March 2022, finding 40 primes.

KPrimeDigits KPrimeDigits KPrimeDigits
285473285473*2^530921+1159829619013619013*2^849281+1255665899449899449*2^981210+1295380
344363344363*2^603009+1181530624511624511*2^962636+1289789923177923177*2^611483+1184081
392479392479*2^958886+1288660653063653063*2^899301+1270723923359923359*2^541446+1162998
420113420113*2^524009+1157749665423665423*2^566441+1170522984173984173*2^872129+1262543
428657428657*2^720223+1216815670309670309*2^520410+1156665989147989147*2^635919+1191437
441923441923*2^774725+1233222680851680851*2^741248+1223144992731992731*2^731740+1220282
481727481727*2^883059+1265833701357701357*2^532979+116044910106931010693*2^560473+1168726
498781498781*2^557856+1167938724351724351*2^675612+120338610136571013657*2^922163+1277605
499729499729*2^725234+1218323735679735679*2^885398+126653810166931016693*2^963829+1290148
504769504769*2^839566+1252741743357743357*2^860491+125904010284311028431*2^556356+1167486
506749506749*2^574746+1173022749971749971*2^843268+125385510451871045187*2^508967+1153221
510893510893*2^556521+1167536750083750083*2^684961+120620010480991048099*2^605090+1182157
609769609769*2^879034+1264622761749761749*2^716354+1215650
615151615151*2^800316+1240925851963851963*2^558637+1168173

No organized testing took place yet for 175 remaining sequences. Some sequences were tested further manually:

• Between July 2023 and September 2023, testing up to 1.5M found 23 primes.

If not specified, current tested depth N = 1.5M.

KPrimeDigitsDepth KPrimeDigitsDepth KPrimeDigitsDepth
272341-2.0M536839-769343769343*2^1230661+1370472Up to prime
273679273679*2^1052058+1316707Up to prime538943-2.0M772411-2.0M
274699-2.0M545401-2.0M777559-2.0M
279361279361*2^1613712+1485782Up to prime545971545971*2^1082956+13260081.103M785153-2.0M
279767-2.0M548033-2.0M794867794867*2^1702787+1512596Up to prime
285601-2.0M553159-2.0M795983-2.0M
286037-2.0M559789559789*2^1030634+1310258Up to prime802613-
287393-2.0M561769-818327-2.0M
289171-566569-819437-2.0M
294181-2.0M571471-2.0M823969-2.0M
305063-580831-826201-2.0M
305147305147*2^1030527+1310226Up to prime583189-2.0M829643-2.0M
310339-2.0M588317-2.0M836687-2.0M
311573-2.0M589021-2.0M843079-2.0M
312121312121*2^1109856+1334106Up to prime590033-2.0M844457-
340441-2.0M599003599003*2^1828141+1550332Up to prime846347-2.0M
340759-599011-2.0M852019852019*2^1763242+1530795Up to prime
351167-2.0M599513599513*2^1282453+1386063Up to prime856043-2.0M
356359-2.0M603767-858079-2.0M
357271357271*2^1370332+1412517Up to prime606199-2.0M860117-2.0M
360331-2.0M609737-867271-
363917363917*2^1655731+1498431Up to prime626303-2.0M870061-2.0M
365221365221*2^1767932+1532207Up to prime630121-2.0M872119-2.0M
365867-632659-878029-2.0M
366953-2.0M632663-2.0M879049879049*2^1174370+1353527Up to prime
368299-633481-2.0M879497-2.0M
381799-2.0M641327-2.0M881537-
382247-2.0M648751-2.0M884723-2.0M
392033-2.0M651857-887153-2.0M
393287-2.0M654499-894409-
400613-2.0M656123-2.0M894827-
412591-2.0M656753-2.0M895579-2.0M
418591-2.0M664639-901067-
433457-2.0M666409666409*2^1083222+1326089Up to prime902191902191*2^1138968+1342870Up to prime
447061447061*2^1206128+1363087Up to prime667861-2.0M902453902453*2^1050893+1316357Up to prime
451351-2.0M674477-2.0M904489-2.0M
452119-2.0M678173-2.0M925907-2.0M
452191-681413-2.0M926371-2.0M
452567-2.0M684617684617*2^1098123+1330574Up to prime935723-
457217-2.0M694973-2.0M946879-
462829-2.0M702707702707*2^1165279+1350790Up to prime957977-2.0M
473543-2.0M703643-961099-2.0M
473567-2.0M705983-964673-2.0M
474323-711833-2.0M968491-2.0M
479783-2.0M714563-971389-
484763-2.0M718849-2.0M972739-
491147-2.0M721141-979039-2.0M
499337-2.0M721397-2.0M983027-2.0M
499561499561*2^1759204+1529579Up to prime734147734147*2^1047447+1315319Up to prime1000313-2.0M
501107501107*2^1058835+1318747Up to prime734177734177*2^1180107+1355254Up to prime10014191001419*2^1675042+1504244Up to prime
502613-2.0M736249-2.0M1004987-
504061504061*2^1714720+1516188Up to prime749447749447*2^1036639+1312066Up to prime1013689-2.0M
515357-2.0M751999751999*2^1589870+1478605Up to prime10348091034809*2^1077230+1324285Up to prime
517913-2.0M757343-2.0M10391271039127*2^1193367+1359246Up to prime
518671518671*2^1157008+1348300Up to prime760583760583*2^1433845+1431637Up to prime1040297-2.0M
520471-766531-2.0M
532703-2.0M766801-

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