Problems & Puzzles: Puzzles

Puzzle 1036. P + R(p) such that...   JM Bergot sent the following nice puzzle:   17+71=88=8*11-->811 is prime and 8+11=19 is prime. 113+311=424=8*53-->853 is prime and 8+53=61 is prime.   Q. Can you find larger emirps factored into the product of one even and one odd number that give two primes in this way?

---

During the week 25-30 April, 2021, contributions came from Giorgos Kalogeropoulos, Paul Cleary, Hakan Summakoglu, Oscar Volpatti, Gennady Gusev.

***

Giorgos wrote:

Since this is "Puzzle 1036", I found the following 100-digit emirp:         

10361036103610361036103610361036103610361036103610361036103610361036103610361036103610
36103610980873         
 

the sum of the emirps can be factored as "even*odd" like this:         

2 * 240849688668866886688668866886688668866886688668866886688668866886688668866886688668
8668866886998587    

So, we get     

1036103610361036103610361036103610361036103610361036103610361036103610361036103610361036103
610980873 + 37808901630163016301630163016301630163016301630163016301630163016301630163016301
63016301630163016301 = 4816993773377337733773377337733773377337733773377337733773377337733773377337733773377337733
773997174 =  2 * 2408496886688668866886688668866886688668866886688668866886688668866886688668866886688668866
886998587 -->  2240849688668866886688668866886688668866886688668866886688668866886688668866886688668866886
6886998587 is prime and       

2 + 2408496886688668866886688668866886688668866886688668866886688668866886688668866886688668
866
886998587 =  2408496886688668866886688668866886688668866886688668866886688668866886688668866886688668866
886998589 is also prime

***

Paul wrote:

Here is a list of emirps with the first prime <=the reverse prime,
duplicates removed and all primes <=179424673. (1000000th prime).
 
3+3 = 6 = 2 * 3-->23 Is Prime and 2 + 3 = 5 Is Prime
11+11 = 22 = 2 * 11-->211 Is Prime and 2 + 11 = 13 Is Prime
17+71 = 88 = 8 * 11-->811 Is Prime and 8 + 11 = 19 Is Prime
107+701 = 808 = 8 * 101-->8101 Is Prime and 8 + 101 = 109 Is Prime
113+311 = 424 = 8 * 53-->853 Is Prime and 8 + 53 = 61 Is Prime
167+761 = 928 = 32 * 29-->3229 Is Prime and 32 + 29 = 61 Is Prime
10067+76001 = 86068 = 4 * 21517-->421517 Is Prime and 4 + 21517 = 21521 Is Prime
11057+75011 = 86068 = 4 * 21517-->421517 Is Prime and 4 + 21517 = 21521 Is Prime
...
5616587+7856165 = 13472752 = 16 * 842047-->16842047 Is Prime and 16 + 842047 = 842063 Is Prime
5620127+7210265 = 12830392 = 8 * 1603799-->81603799 Is Prime and 8 + 1603799 = 1603807 Is Prime
5620469+9640265 = 15260734 = 2 * 7630367-->27630367 Is Prime and 2 + 7630367 = 7630369 Is Prime
5626631+1366265 = 6992896 = 1024 * 6829-->10246829 Is Prime and 1024 + 6829 = 7853 Is Prime
5626853+3586265 = 9213118 = 2 * 4606559-->24606559 Is Prime and 2 + 4606559 = 4606561 Is Prime

***

Hakan wrote:

I found 49170 different emirps such that for <10^8. This file contains complete list of these 49170 emirps.

***

Oscar wrote:

if I correctly understood puzzle 1036, we are asked to find numbers x which can be written:

as a sum p+q, where (p,q) is an emirp pair;

as an even/odd product a*b, where both the sum a+b and the concatenation a.b are prime numbers too.

The first such numbers are the examples 88 and 424, next number is x = 808:

808 = 107+701, 107 and 701 are an emirp pair;

808 = 8*101, 109 and 8101 are prime numbers.

 

An interesting solution is x = 1293292 = 4*7*11*13*17*19;

as p+q in 11 ways: 

307589 + 985703

313979 + 979313

314879 + 978413

315779 + 977513

324869 + 968423

333959 + 959333

354839 + 938453

364829 + 928463

383909 + 909383

385709 + 907583

387509 + 905783

as a*b in 10 ways: 

99484 * 13

14212 * 91

6916 * 187

5236 * 247

1292 * 1001

988 * 1309

884 * 1463

572 * 2261

532 * 2431

68 * 19019

 

Below 2*10^8, the product-champion is x = 124522552 = 8*7*11^2*17*23*47;

as p+q in 2 ways: 

33320219 + 91202333

33511019 + 91011533

as a*b in 15 ways:

124522552 * 1

7324856 * 17

1046408 * 119

318472 * 391

147016 * 847

115192 * 1081

21896 * 5687

16456 * 7567

6776 * 18377

2632 * 47311

1288 * 96679

968 * 128639

376 * 331177

184 * 676753

56 * 2223617

  
 

Below 2*10^8, the sum-champion is x = 107999980;

as a*b in 2 ways:

14740 * 7327

220 * 490909

as p+q in 100 ways: 

10227779 + 97772201

10345679 + 97654301

10818179 + 97181801

10872179 + 97127801

10890179 + 97109801

10927079 + 97072901

11036969 + 96963011

11072969 + 96927011

11218769 + 96781211

11236769 + 96763211

11654369 + 96345611

11754269 + 96245711

11854169 + 96145811

11918069 + 96081911

11945069 + 96054911

11963069 + 96036911

12154859 + 95845121

12190859 + 95809121

12481559 + 95518421

12745259 + 95254721

12990059 + 95009921

13054949 + 94945031

13318649 + 94681331

13372649 + 94627331

13863149 + 94136831

14136839 + 93863141

14254739 + 93745241

14536439 + 93463541

14545439 + 93454541

15054929 + 92945051

15163829 + 92836151

15245729 + 92754251

15327629 + 92672351

15409529 + 92590451

15745229 + 92254751

15863129 + 92136851

15927029 + 92072951

15945029 + 92054951

16045919 + 91954061

16181819 + 91818161

16354619 + 91645361

16363619 + 91636361

16563419 + 91436561

16709219 + 91290761

17263709 + 90736271

17272709 + 90727271

17518409 + 90481571

17554409 + 90445571

17909009 + 90090971

17963009 + 90036971

30018977 + 77981003

30672377 + 77327603

30972077 + 77027903

30990077 + 77009903

31090967 + 76909013

31436567 + 76563413

31490567 + 76509413

31518467 + 76481513

31790267 + 76209713

32018957 + 75981023

32118857 + 75881123

32209757 + 75790223

32427557 + 75572423

32872157 + 75127823

32945057 + 75054923

32954057 + 75045923

33072947 + 74927033

33090947 + 74909033

33163847 + 74836133

33490547 + 74509433

33572447 + 74427533

33636347 + 74363633

33645347 + 74354633

33754247 + 74245733

34018937 + 73981043

34381637 + 73618343

34636337 + 73363643

34845137 + 73154843

34927037 + 73072943

35036927 + 72963053

35072927 + 72927053

35145827 + 72854153

35290727 + 72709253

35318627 + 72681353

35381627 + 72618353

35427527 + 72572453

35454527 + 72545453

35472527 + 72527453

35545427 + 72454553

36045917 + 71954063

36063917 + 71936063

36136817 + 71863163

36281717 + 71718263

36381617 + 71618363

36445517 + 71554463

36527417 + 71472563

36727217 + 71272763

36918017 + 71081963

36954017 + 71045963

37527407 + 70472573

 

A 100-digit solution:

x = 2*10^99 + 61056105*10^46 + 2

p = 10^99 + 27350733*10^46 + 1

q = 10^99 + 33705372*10^46 + 1

a = 2

b = 10^99 + 305280525*10^45 + 1

***

Gennady wrote:

My solution:

prime=1470289*10^80+1=1470289000000000000000000000000000000000000000000
00000000000000000000000000000000000001;

emirp=10^86+9820741=100000000000000000000000000000000000000000000000000
000000000000000000000000000009820741;

prime+emirp=247028900000000000000000000000000000000000000000000000000000
000000000000000000009820742=f1*f2;

f1=2062225326261701868482;

f2=119787540601977543943517177853617481897244818680202369226260652931;

f1 || f2 = 20622253262617018684821197875406019775439435171778536174818972448186802
02369226260652931 - prime

f1 + f2 = 119787540601977543943517177853617481897244820742427695487962521413 - prime

***