Problems & Puzzles: Puzzles

Puzzle 23.- Pal-primes adding consecutive primes

Question No. 1 : Find a Pal-prime adding consecutive primes.

The least case  (for a pal-prime of more than 2 digits) is :

41+43+47 =131

Jud McCranie is the author of the current record (August 26, 1998):

4272827261 + 4272827263 + 4272827297 = 12818481821

Can you get the least case for 5, 7, & 9 digits ? Can you get a larger one than the McCranie record , that is to say can you get the least case for 13 digits?

Question No. 2 : Find a Pal-prime adding consecutive pal- primes.

The least example (using pal-primes of more than 2 digits) is :

101 + 131 + 151 = 383

Carlos Rivera has the current record (191 digits):

1(0)87132298010892231(0)871 +

1(0)87132300858003231(0)871 +

1(0)87132301111103231(0)871 =

3(0)87396899979998693(0)873

Can you find a larger one.

See both records at the Patrick De Geest pages about Pal-primes: http://www.worldofnumbers.com/palpri.htm

Solution

Question a)

Jud McCranie has gotten (October 5, 1998) new and higger pal-primes adding consecutive primes.  Here are his last records:

The smallest with 5 digits:
10501 = 3491 + 3499 + 3511

The smallest with 7 digits:
1126211 = 375391 + 375407 + 375413

The smallest with 9 digits:
100404001 = 33467981 + 33467989 + 33468031

The smallest with 11 digits:
10021912001 = 3340637303 + 3340637347 + 3340637351

The largest 11-digit pal-prime that is the sum of 3 consecutive
primes.

99988988999 = 33329662973 + 33329662999 + 33329663027

The smallest with 13 digits:
1000051500001 = 333350499973 + 333350499991 + 333350500037

The largest 13-digit pal-prime that is the sum of 3 consecutive
primes:

9999656569999 = 3333218856647 + 3333218856673 + 3333218856679

The smallest 15-digit pal-prime that is the sum of 3 consecutive primes:
100000929000001 = 33333642999977 + 33333642999991 + 33333643000033

The two largest 15-digit pal-primes that are the sum of 3
consecutive primes:

999998727899999 = 333332909299937 + 333332909300029 + 333332909300033

999998373899999 = 333332791299943 + 333332791300013 + 333332791300043

***

J. K. Andersen got (as usual) really big solutions to both questions... (May 2003):

Question No. 1

For puzzle 7, I found a 527-digit pal-prime which was the sum of 3 consecutive primes where the middle p was also palindromic: Let w be "0" followed by 124 concatenations of "70". p is the concatenation 1w728092807290919092708290827w1. The equation is (p-180) + (p) + (p+200) = (3p+20). The 4 primes were proved with Marcel Martin's Primo.

Question No. 2

Let z be 243 repetitions of 0.

Then a 515-digit solution is:

1z101031223000000000322130101z1 + 1z101031223000313000322130101z1 + 1z101031223008545800322130101z1 = 3z303093669008858800966390303z1

The sum of 3 palindromes with the same decimal length becomes a palindrome if there are no carries in the addition. My strategy: Find pal-primes p with several 0's in the middle and no digits above 3. For each such p, find the next 2 pal-primes and test whether the sum of all 3 is a pal-prime. The 0's in the middle of p means the sum of only 2 (instead of 3) numbers must be without carries, improving the palindrome odds a lot.

A prp solution was found with a C program using Michael Scott's Miracl bigint library. PrimeForm/GW proved the 3 consecutive pal-primes and Primo proved the sum.

***

One week later J. K. Andersen got this:

Question No. 1

Three titanic solutions:

p = 10^1000+76245954267*10^495+1

= (x-2846) + (x+550) + (x+2298), where x = (p-2)/3

p = 10^1000+1308107018031*10^494+1

= (x-3772) + (x-2094) + (x+5868), where x = (p-2)/3

p = 10^1000+1576219126751*10^494+1

= (x-1763) + (x-187) + (x+1951), where x = (p-1)/3

My strategy: Find pal-primes p and for each, find the primes a and b closest before and after p/3. Test whether p-a-b is prime, if so then test if it is consecutive with a and b to give a solution I trial factored with my own C program. Prp-tests were performed with the GMP library and PrimeForm/GW. The 3 pal-primes were proved with PrimeForm/GW and all other primes with Marcel Martin's Primo. This is probably the 3 smallest titanic solutions. All smaller pal-primes have been eliminated, but only by finding probable primes a and b, giving composite p-a-b or a non-consecutive prime in a single case.

***

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