## Prime numbers P : (P+1)^2 + 1

Find prime numbers   $P$   such that

$(P + 1)^2 \; + \; 1 \; = \; Q$    is a prime number

$Q \; + \; DigitReversal \,( \,Q \,) \; = \; r$    is a palindrome

$Q^2 \; + \; DigitReversal \,( \,Q^2 \,) \; = \; R$    is a palindrome

The first few prime solutions:

Do the prime solutions always end in digit 9?

Paul found :

$(P - 1)^2 \; + \; 1 \; = \; Q$    is a prime number

$(P - 3)^2 \; + \; 1 \; = \; Q$    is a prime number

$(P + 3)^2 \; + \; 1 \; = \; Q$    is a prime number

math grad - Interest: Number theory
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### 5 Responses to Prime numbers P : (P+1)^2 + 1

1. Paul says:

Here are solutions with P ,=1000000, Format is {p, q, r, R}
and it does look like it always ends with a 9.

19 , 401 , 505 , 268862
89 , 8101 , 9119 , 75888857
109 , 12101 , 22222 , 248868842
1549 , 2402501 , 3454543 , 6777512157776
4909 , 24108101 , 34288243 , 683828868828386
7489 , 56100101 , 66200266 , 4167343663437614
8999 , 81000001 , 91000019 , 7561002772001657
13999 , 196000001 , 296000692 , 48416029692061484
32999 , 1089000001 , 2089009802 , 2185929714179295812
159899 , 25568010001 , 35569096553 , 753743767525767347357
201499 , 40602250001 , 50607470605 , 2648596778558776958462
214499 , 46010250001 , 56015451065 , 3116993359669533996113
379999 , 144400000001 , 244400004442 , 30851360888488806315803
409999 , 168100000001 , 268100001862 , 38257610263636201675283
419999 , 176400000001 , 276400004672 , 41116960825652806961114
650099 , 422630010001 , 522640046225 , 278636188897798881636872
724499 , 524900250001 , 624952259426 , 375525275662266572525573
764999 , 585225000001 , 685225522586 , 442488354697796453884244
1239899 , 1537352010001 , 2537454547352 , 3363655286768676825563633
1409999 , 1988100000001 , 2988100018892 , 4952541636796976361452594
2248999 , 5058001000001 , 6058002008506 , 35583394177122177149338553
2309999 , 5336100000001 , 6336100016336 , 38473963437611673436937483
2939899 , 8643012010001 , 9643114113469 , 84703698767533576789630748
3198799 , 10232321440001 , 20236733763202 , 204788644882909288446887402
3349999 , 11222500000001 , 21222500522212 , 225944506794242497605449522
4979999 , 24800400000001 , 34800400400843 , 715059848166989661848950517
5989999 , 35880100000001 , 45880100108854 , 2287381596681771866951837822
6501799 , 42273403240001 , 52277633677225 , 2787886665898998985666887872

Paul.

2. Paul says:

Here are solutions when it is (p-1)^2 + 1

11 , 101 , 202 , 20402
41 , 1601 , 2662 , 3586853
401 , 160001 , 260062 , 35602620653
701 , 490001 , 590095 , 340189981043
8101 , 65610001 , 75611657 , 5304893553984035
9001 , 81000001 , 91000019 , 7561002772001657
19001 , 361000001 , 461000164 , 230321227722123032
20101 , 404010001 , 504020405 , 263244889988442362
28001 , 784000001 , 884000488 , 714656866668656417
69001 , 4761000001 , 5761001675 , 32667143599534176623
70501 , 4970250001 , 5970770795 , 34703889499498830743
81001 , 6561000001 , 7561001657 , 53046743144134764035
115001 , 13225000001 , 23225052232 , 274900679646976009472
139501 , 19460250001 , 29465456492 , 478706354202453607874
280001 , 78400000001 , 88400000488 , 7146560086666800656417
319001 , 101761000001 , 201761167102 , 20355323651415632355302
325001 , 105625000001 , 205625526502 , 21156645836463854665112
361001 , 130321000001 , 230321123032 , 26983587647474678538962
420001 , 176400000001 , 276400004672 , 41116960825652806961114
700001 , 490000000001 , 590000000095 , 340100000089980000001043
751001 , 564001000001 , 664001100466 , 418097328823328823790814
880001 , 774400000001 , 874400004478 , 699695368846648863596996
1810001 , 3276100000001 , 4276100016724 , 20732831435566553413823702
2890001 , 8352100000001 , 9352100012539 , 79757574650777705647575797
4900001 , 24010000000001 , 34010000001043 , 676480100020888020001084676
4997101 , 24971008410001 , 34972488427943 , 723579872088383880278975327
6990001 , 48860100000001 , 58860100106885 , 3387309392287997822939037833
7900001 , 62410000000001 , 72410000001427 , 4895008100284334820018005984
11200001 , 125440000000001 , 225440000044522 , 25735193608805450880639153752
11800001 , 139240000000001 , 239240000042932 , 29387777608487478480677778392
12490001 , 156000100000001 , 256000101000652 , 34336031400031613000413063343
13990001 , 195720100000001 , 295720101027592 , 48306357748429692484775360384
15300001 , 234090000000001 , 334090000090433 , 64798128108186868180182189746
15455101 , 238860116010001 , 338870727078833 , 67056178324374247342387165076

Paul.

• benvitalis says:

You beat me to it.
How about (p – 3)^2 + 1 and (p + 3)^2 + 1

3. Paul says:

(p-3)^2+1 is

13 , 101 , 202 , 20402
23 , 401 , 505 , 268862
43 , 1601 , 2662 , 3586853
113 , 12101 , 22222 , 248868842
1553 , 2402501 , 3454543 , 6777512157776
4603 , 21160001 , 31166113 , 547768888867745
49003 , 2401000001 , 3401001043 , 6764803088803084676
71503 , 5112250001 , 6112772116 , 36135642799724653163
89003 , 7921000001 , 8921001298 , 72742265866856224727
230003 , 52900000001 , 62900000926 , 3798410085115800148973
250403 , 62700160001 , 72706260727 , 4931540065665600451394
251003 , 63001000001 , 73001010037 , 4969128007337008219694
260003 , 67600000001 , 77600000677 , 5569760025445200679655
275503 , 75900250001 , 85905450958 , 6760897984334897980676
284003 , 80656000001 , 90656065609 , 7505392467777642935057
361003 , 130321000001 , 230321123032 , 26983587647474678538962
455003 , 207025000001 , 307025520703 , 52859355666866655395825
754003 , 568516000001 , 668516615866 , 423210672988889276012324
890003 , 792100000001 , 892100001298 , 727422412486684214224727
1090003 , 1188100000001 , 2188100018812 , 2411581636734376361851142
1460003 , 2131600000001 , 3131600061313 , 5543718583628263858173455
2090003 , 4368100000001 , 5368100018635 , 29080297873788737879208092
2890003 , 8352100000001 , 9352100012539 , 79757574650777705647575797
4244003 , 18011536000001 , 28011599511082 , 424415699399565993996514424
4900003 , 24010000000001 , 34010000001043 , 676480100020888020001084676
5006003 , 25060036000001 , 35060099006053 , 728005674342989243476500827
6500003 , 42250000000001 , 52250000005225 , 2785062500054884500052605872
7657103 , 58631180410001 , 68632588523686 , 4437895956575885756595987344
8373983 , 70123541040401 , 80527555572508 , 5997735159677887769515377995
14180003 , 201072400000001 , 301072404270103 , 50430110885880808858801103405
14900003 , 222010000000001 , 322010000010223 , 59288440102044844020104488295

and (p+3)^2+1 is

7 , 101 , 202 , 20402
17 , 401 , 505 , 268862
37 , 1601 , 2662 , 3586853
107 , 12101 , 22222 , 248868842
397 , 160001 , 260062 , 35602620653
4597 , 21160001 , 31166113 , 547768888867745
7487 , 56100101 , 66200266 , 4167343663437614
13997 , 196000001 , 296000692 , 48416029692061484
25097 , 630010001 , 730020037 , 496932664466239694
27997 , 784000001 , 884000488 , 714656866668656417
88997 , 7921000001 , 8921001298 , 72742265866856224727
114997 , 13225000001 , 23225052232 , 274900679646976009472
124897 , 15600010001 , 25601010652 , 343380315161513083343
148997 , 22201000001 , 32201010223 , 592884605484506488295
201497 , 40602250001 , 50607470605 , 2648596778558776958462
297097 , 88268410001 , 98269896289 , 8791598568118658951978
324997 , 105625000001 , 205625526502 , 21156645836463854665112
379997 , 144400000001 , 244400004442 , 30851360888488806315803
393977 , 155220240401 , 259262262952 , 34897946274247264979843
639997 , 409600000001 , 509600006905 , 267772162918819261277762
889997 , 792100000001 , 892100001298 , 727422412486684214224727
1839997 , 3385600000001 , 4385600065834 , 21462287577766777578226412
4243997 , 18011536000001 , 28011599511082 , 424415699399565993996514424
4899997 , 24010000000001 , 34010000001043 , 676480100020888020001084676
6501797 , 42273403240001 , 52277633677225 , 2787886665898998985666887872
9199997 , 84640000000001 , 94640000004649 , 8163929600829779280069293618
11299997 , 127690000000001 , 227690000096722 , 26304736108355455380163740362
12199997 , 148840000000001 , 248840000048842 , 32153345608679497680654335123
12310997 , 151560721000001 , 251560848065152 , 32970676562184248126567607923
12999997 , 169000000000001 , 269000000000962 , 38561000000083638000000016583
14188997 , 201327721000001 , 301327848723103 , 50532875698688488689657823505

Paul.