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

***

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

***

 Records   |  Conjectures  |  Problems  |  Puzzles

 Home | Melancholia | Problems & Puzzles | References | News | Personal Page | Puzzlers | Search | Bulletin Board | Chat | Random Link Copyright © 1999-2012 primepuzzles.net. All rights reserved.