Real Riesel conjectures and proofs

Started: Mar. 24, 2024
Last update: Apr. 21, 2026

Compiled by Gary Barnes based on work completed and coordinated by Riley Fisher.

Real Riesel conjecture reservations

Abbreviations:  b = base, MOB = multiple of the base

Real Riesel conjectures are the same as the classical Riesel conjectures except that primes must be b^n > k.

For Riesel base 2 k<=762701 only, the following will apply:
1. Only k's that have a small prime where b^n <= k and n > 1 will have their primes shown and are tested with the Real Riesel project.
2. k's that do not have a prime where b^n <= k and n > 1 will not have their primes shown and are worked on by other projects.
   The k's must still be accounted for and are counted and shown separately.

Requirements and inclusions:
1. Conjectures must have a finite covering set and cannot be a MOB.
2. Primes must be b^n > k.
3. k-values that are a MOB are included in the conjectures but some may be excluded.  See exclusion 3 below.

k-values will be excluded if any of the following conditions are met:
1. All n-values are covered by one trivial factor.
2. All n-values are covered by algebraic factors or a combination of algebraic and trivial factor(s).
3. k is a MOB and both k*b^(n-1)-1 and k*b^n-1 are composite, where n = floor [ logb(k) ]. They will have the same prime as k / b.

Testing not done through other projects is coordinated at Mersenneforum Real Riesel base 2 testing/reservations.

Base Conjectured Riesel k Covering set k's that make a full covering set with all or partial algebraic factors Trivial k's (factor) Remaining k's to find prime
(n testing limit)		 Top 10 k's with largest first primes: k (n) Comments / accounting of all k's
2 509203 3, 5, 7, 13, 17, 241   none 63 k's remaining at n>=2.2M.

See k's and test limits at Real Riesel Base 2 remain.

Breakdown:

Real Riesel:
19 k's remaining at n=2.2M.

Original base 2 project
(k's will not be tested by Real Riesel):
41 k's remaining at n=16.8M.

Original base 2 1st/2nd conjectures even k efforts
(k's will not be tested by Real Riesel):
3 k's remaining at n>=9M.		 Real Riesel:
64016 (2093282)
17579 (1712236)
327643 (1656139)
200683 (1509271)
147911 (1233334)
429209 (849392)
7933 (820795)
493931 (742430)
493693 (552827)
305999 (538088)		 Original project k's are worked on by PrimeGrid's Riesel Problem  project.

Original even k efforts are worked on by the Conjectures 'R Us project.

all-ks-real-riesel-base2.zip
2
2nd conjecture		 762701 3, 5, 7, 13, 17, 241   none 34 k's remaining at n>=2.2M.

See k's and test limits at Real Riesel 2nd Base 2 remain.

Breakdown:

Real Riesel:
12 k's remaining at n=2.2M.

Original base 2 project
(k's will not be tested by Real Riesel):
22 k's remaining at n=9M.

 		 Real Riesel:
542383 (2149807)
662257 (1574497)
737713 (1454583)
710249 (1258016)
586753 (1038895)
565303 (841475)
711701 (760446)
756116 (706602)
573569 (650240)
758557 (647409)		 Only 509203<k<762701 are considered.

Original project k's are worked on by the Conjectures 'R Us project.

all-ks-real-riesel-base2-2nd-conj.zip
2
3rd conjecture		 777149 3, 5, 7, 13, 19, 37, 73   none 765389 (3M)
768119 (3M)
776351 (3M)		 772897 (1133789)
769717 (716129)
771253 (291187)
771269 (263112)
766777 (217965)
774223 (175407)
763469 (136424)
768307 (134333)
774419 (95988)
775933 (84843)		 Only 762701<k<777149 are considered.

all-ks-real-riesel-base2-3rd-conj.txt
2
4th conjecture		 790841 3, 5, 7, 13, 19, 37, 73   none 787663 (3M) 780427 (2082633)
785647 (441877)
783073 (402635)
780809 (296392)
783217 (185893)
784109 (99884)
781573 (54247)
785653 (45647)
784781 (39150)
779503 (21243)		 Only 777149<k<790841 are considered.

all-ks-real-riesel-base2-4th-conj.txt
2
5th conjecture		 992077 3, 5, 7, 13, 17, 241   none 37 k's remaining at n>=1M.

See k's and test limits at Real Riesel 5th Base 2 remain.		 981493 (3266031)
866107 (710057)
985559 (688036)
989741 (655790)
841243 (624815)
851407 (582489)
804187 (472817)
906221 (449630)
826679 (419892)
947038 (414739)		 Only 790841<k<992077 are considered.

all-ks-real-riesel-base2-5th-conj.zip
2
6th conjecture		 1106681 3, 5, 7, 13, 19, 37, 73   none 998321 (1M)
1010759 (1M)
1032431 (1M)
1033138 (1M)
1035269 (1M)
1047763 (1M)
1048273 (1M)
1049917 (1M)
1051859 (1M)
1054811 (1M)
1064561 (1M)
1069469 (1M)
1072031 (1M)
1080179 (1M)
1081109 (1M)
1082561 (1M)
1087559 (1M)
1090259 (1M)
1093087 (1M)
1093271 (1M)
1099751 (1M)		 1024837 (983965)
1030709 (906496)
1078841 (858154)
1095329 (849764)
999041 (842606)
1103173 (747203)
1084499 (603860)
1033769 (577800)
1088021 (558914)
1017937 (525993)		 Only 992077<k<1106681 are considered.

all-ks-real-riesel-base2-6th-conj.zip

Contributors:
gd_barnes
Happy5214
Hibou
ikari
pinhodecarlos
Rytis