![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sgprmdvdsmersenne | Structured version Visualization version GIF version |
Description: If 𝑃 is a Sophie Germain prime (i.e. 𝑄 = ((2 · 𝑃) + 1) is also prime) with 𝑃≡3 (mod 4), then 𝑄 divides the 𝑃-th Mersenne number MP. (Contributed by AV, 20-Aug-2021.) |
Ref | Expression |
---|---|
sgprmdvdsmersenne | ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) ∧ (𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ)) → 𝑄 ∥ ((2↑𝑃) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 744 | . 2 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) ∧ (𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ)) → 𝑃 ∈ ℙ) | |
2 | simprr 750 | . 2 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) ∧ (𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ)) → 𝑄 ∈ ℙ) | |
3 | oveq1 6801 | . . . 4 ⊢ (𝑄 = ((2 · 𝑃) + 1) → (𝑄 mod 8) = (((2 · 𝑃) + 1) mod 8)) | |
4 | 3 | adantr 466 | . . 3 ⊢ ((𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ) → (𝑄 mod 8) = (((2 · 𝑃) + 1) mod 8)) |
5 | prmz 15597 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
6 | mod42tp1mod8 42048 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ (𝑃 mod 4) = 3) → (((2 · 𝑃) + 1) mod 8) = 7) | |
7 | 5, 6 | sylan 563 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) → (((2 · 𝑃) + 1) mod 8) = 7) |
8 | 4, 7 | sylan9eqr 2827 | . 2 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) ∧ (𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ)) → (𝑄 mod 8) = 7) |
9 | simprl 748 | . 2 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) ∧ (𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ)) → 𝑄 = ((2 · 𝑃) + 1)) | |
10 | sfprmdvdsmersenne 42049 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑄 ∈ ℙ ∧ (𝑄 mod 8) = 7 ∧ 𝑄 = ((2 · 𝑃) + 1))) → 𝑄 ∥ ((2↑𝑃) − 1)) | |
11 | 1, 2, 8, 9, 10 | syl13anc 1478 | 1 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) ∧ (𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ)) → 𝑄 ∥ ((2↑𝑃) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4787 (class class class)co 6794 1c1 10140 + caddc 10142 · cmul 10144 − cmin 10469 2c2 11273 3c3 11274 4c4 11275 7c7 11278 8c8 11279 ℤcz 11580 mod cmo 12877 ↑cexp 13068 ∥ cdvds 15190 ℙcprime 15593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-inf2 8703 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 ax-pre-sup 10217 ax-addf 10218 ax-mulf 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-isom 6041 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-of 7045 df-ofr 7046 df-om 7214 df-1st 7316 df-2nd 7317 df-supp 7448 df-tpos 7505 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-2o 7715 df-oadd 7718 df-er 7897 df-ec 7899 df-qs 7903 df-map 8012 df-pm 8013 df-ixp 8064 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-fsupp 8433 df-sup 8505 df-inf 8506 df-oi 8572 df-card 8966 df-cda 9193 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-div 10888 df-nn 11224 df-2 11282 df-3 11283 df-4 11284 df-5 11285 df-6 11286 df-7 11287 df-8 11288 df-9 11289 df-n0 11496 df-xnn0 11567 df-z 11581 df-dec 11697 df-uz 11890 df-q 11993 df-rp 12037 df-ioo 12385 df-ico 12387 df-fz 12535 df-fzo 12675 df-fl 12802 df-mod 12878 df-seq 13010 df-exp 13069 df-fac 13266 df-hash 13323 df-cj 14048 df-re 14049 df-im 14050 df-sqrt 14184 df-abs 14185 df-clim 14428 df-prod 14844 df-dvds 15191 df-gcd 15426 df-prm 15594 df-phi 15679 df-pc 15750 df-struct 16067 df-ndx 16068 df-slot 16069 df-base 16071 df-sets 16072 df-ress 16073 df-plusg 16163 df-mulr 16164 df-starv 16165 df-sca 16166 df-vsca 16167 df-ip 16168 df-tset 16169 df-ple 16170 df-ds 16173 df-unif 16174 df-hom 16175 df-cco 16176 df-0g 16311 df-gsum 16312 df-prds 16317 df-pws 16319 df-imas 16377 df-qus 16378 df-mre 16455 df-mrc 16456 df-acs 16458 df-mgm 17451 df-sgrp 17493 df-mnd 17504 df-mhm 17544 df-submnd 17545 df-grp 17634 df-minusg 17635 df-sbg 17636 df-mulg 17750 df-subg 17800 df-nsg 17801 df-eqg 17802 df-ghm 17867 df-cntz 17958 df-cmn 18403 df-abl 18404 df-mgp 18699 df-ur 18711 df-srg 18715 df-ring 18758 df-cring 18759 df-oppr 18832 df-dvdsr 18850 df-unit 18851 df-invr 18881 df-dvr 18892 df-rnghom 18926 df-drng 18960 df-field 18961 df-subrg 18989 df-lmod 19076 df-lss 19144 df-lsp 19186 df-sra 19388 df-rgmod 19389 df-lidl 19390 df-rsp 19391 df-2idl 19448 df-nzr 19474 df-rlreg 19499 df-domn 19500 df-idom 19501 df-assa 19528 df-asp 19529 df-ascl 19530 df-psr 19572 df-mvr 19573 df-mpl 19574 df-opsr 19576 df-evls 19722 df-evl 19723 df-psr1 19766 df-vr1 19767 df-ply1 19768 df-coe1 19769 df-evl1 19897 df-cnfld 19963 df-zring 20035 df-zrh 20068 df-zn 20071 df-mdeg 24036 df-deg1 24037 df-mon1 24111 df-uc1p 24112 df-q1p 24113 df-r1p 24114 df-lgs 25242 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |