Problems & Puzzles: Puzzles

Puzzle 495. P*R(P)+1 = Prime JM Bergot sent the following puzzle: 61*16+1=977 (prime) Q. Find larger solutions.

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Contributions came from W.E. Clark, Jan van Delden, Qushunliang, faideh Firoozbakht, JC Rosa, J.K.Andersen & Jasper Duba.

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W.E. Clark wrote:

In puzzle 495, I assume that R(p) means the number obtained by reversing
the digits of p. If so, here's a 500 digit prime p such that
p*R(p)+1 is a 999 digit prime:

2404239999301527981560311338889153161224483626188\
9446801075349696292603139491179868024797543542498\
2194446945697125845031221688710611059759445449285\
6111058118952214499647722622228341524632383594851\
3657700328642692064973780163601322970001193833409\
6064306703235838593881907526825217741403507252544\
2518109941032646149439974246095805586554978256657\
0096986785585740578743280499554694477442665327301\
9553931802401369049209658409414843990273942245488\
8929632338612106869581137490099331014882877882063\
8108526243

(Found in 146 random trials and using Maple's probabilistic primality
function "isprime" to check primality.)

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Jan van Delden wrote:

My contribution to puzzle 495:

[number of digits d, number of solutions n]
[1,1] (p=2),
[2,1] (p=61)
[3,16]
[4,67]
[5,512]
[6,2906]
[7,25896]
[8,163598]

No solution p*r(p)+1 I found is a palindrom.

The revised problem p*r(p )-1=q, has no solutions, since now 3|q
So (p*r(p)-1)/3=q could be examined instead, which has the palprime solution: p=41, q=191.

Observe that p=r(p) mod 3, so p*r(p )-1 mod 3 = p*p-1 mod 3 =(p-1) (p+1) mod 3,
which is equal to 0 mod 3, since p is either 1 mod 3 or 2 mod 3.

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Qushunliang wrote:

P=21774393094431644344880796528717836642571400508502821467973.
R(P)=37976412820580500417524663871782569708844344613449039347712=2^147 * 7^13 * 13^3.
P*R(P)+1=82691334107173341319527530997185069936687924628447064369516451941431499024145934/
0945750613738392166641139186218827777 is a prime.

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Farideh wrote:

There exist infinitely many such primes.

These primes up to 5000 are :2,61,211,223,251,269,283,443,463,479,499,631,647,677,827,839,877,2029,2053, 2131,2293,2339,2371,2389,2411,2459,2549,2633,2699,2713,2753,2791,2819,4007,
4099,4133,4229,4253,4273,4297,4409,4457,4483,4507,4547,4567,4639,4723,4759,
4789,4799,4889,4909,4937,4999

I also found many large such primes. Two of them that has two or three distinct digits including
1 & 6 are p1 & p2. p1 has 121 digits and p2 has 895 digits as follows.

p1 = 6(120).1
= 666666666666666666666666666666666666666666666666666666666666
6666666666666666666666666666666666666666666666666666666666661

p2 = 6(43).0(7).1(845)

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J.C. Rosa wrote:

My best solution is :

899999999999801*108999999999998+1=98099999999976509000000000399

I have a question:  For P*R(P)+1 is it possible to find a palprime ?

I have tested P up to 2447338141 and I have found only two palindromic numbers:
23*32+1=737 and 41*14+1=575

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J. K. Andersen wrote:

a) There are 7 consecutive primes P [of the asked ones] from 8682559 to 8682691.

b) I tested a lot of large known primes and found a solution for P = 158841*2^11111+1 from the Prime Pages database: http://primes.utm.edu/primes/page.php?id=29133. P has 3350 digits and was found in 1997 by ... Carlos Rivera! PrimeForm/GW found and proved P*R(P)+1.

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Jasper Duba wrote:

I wasn't sure but I assumed that R(P) means the reverse of P, i hope that was right. Here are a handful of my 'large' solutions

...

P is 9389605607969712810096871871461284488013983297494483681514902480158722964501007440/
30184473209589251325876215793346231226359122913703478105387279153636740493577671603904/
726586048066574474007365114654879697353895897272171728479159942

RP is 249951974827171272798598353796978456411563700474475660840685627409306176775394047/
63635197278350187430731922195362213264339751267852315298590237448103044700105469227851/
0842094151863844947923893108844821641781786900182179697065069839


prime is
2346950464560311870917964519426835979271753140683510236155867115625488398634896281498/
2670146681351121811955232447189669757972607145470867898443610450288636982800834719142/
0627338286139607925885761614514037990036836133153034117454552775803992882025922949950/
3649830430077443110193616429397530875804138168940104840847440559637326325271627786786/
1777548674741785617237245169784707499638326893820783205745175413208564110546144237368/
3183043091744920182600588226281189339

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