Problems & Puzzles:
Puzzles
Puzzle 23. Palprimes adding consecutive primes
Question No. 1 : Find a Palprime adding consecutive primes.
The least case (for a palprime 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 Palprime adding consecutive pal primes.
The least example (using palprimes of more than 2 digits) is :
101 + 131 + 151 = 383
Carlos Rivera has the current record (191 digits):
1(0)_{87}132298010892231(0)_{87}1 +
1(0)_{87}132300858003231(0)_{87}1 +
1(0)_{87}132301111103231(0)_{87}1 =
3(0)_{87}396899979998693(0)_{87}3
Can you find a larger one.
See both records at the Patrick De Geest pages about Palprimes:
http://www.worldofnumbers.com/palpri.htm
Solution
Question a)
Jud McCranie has gotten (October 5, 1998) new and
higger palprimes 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 11digit palprime that
is the sum of 3 consecutive
primes.
99988988999 = 33329662973 + 33329662999
+ 33329663027
The smallest with 13 digits:
1000051500001 = 333350499973 + 333350499991
+ 333350500037
The largest 13digit palprime that
is the sum of 3 consecutive
primes:
9999656569999 = 3333218856647 + 3333218856673
+ 3333218856679
The smallest 15digit palprime that is the sum of 3 consecutive primes:
100000929000001 = 33333642999977 + 33333642999991 + 33333643000033
The two largest 15digit palprimes
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 527digit palprime 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 (p180) + (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 515digit 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
palprimes p with several 0's in the middle and no digits above 3. For
each such p, find the next 2 palprimes and test whether the sum of all 3
is a palprime. 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 palprimes 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
= (x2846) + (x+550) + (x+2298), where x = (p2)/3
p = 10^1000+1308107018031*10^494+1
= (x3772) + (x2094) + (x+5868), where x = (p2)/3
p = 10^1000+1576219126751*10^494+1
= (x1763) + (x187) + (x+1951), where x = (p1)/3
My strategy: Find palprimes p and for each, find
the primes a and b closest before and after p/3. Test whether pab 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. Prptests were performed
with the GMP library and PrimeForm/GW. The 3 palprimes were proved with
PrimeForm/GW and all other primes with Marcel Martin's Primo. This is
probably the 3 smallest titanic solutions. All smaller palprimes have
been eliminated, but only by finding probable primes a and b, giving
composite pab or a nonconsecutive prime in a single case.
***
