Problems & Puzzles: Puzzles

Puzzle 705 2213 2213 is the smallest prime P such that it can be partitioned in three parts A, B, C -not necessarily of the same length- such that: 1) P=A&B&C 2) P=A^3+B^3+C^3 2213=2^3+2^3+13^3 From our past puzzles 104 & 385 we know that the next prime like 2213 is the prime 333667001. BTW, this second prime is part of an infinite family of integer solutions (3)n(6)n-17(0)n-11 that Giovanni Resta has proved that is prime only for n=3, 4 & 46,...? Q. ¿Are there more solutions like 2213 and out of the Resta's infinite family solutions?

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Giovanni wrote:

Besides 2213 and the solutions listed in Puzzle 104, i.e.,

8740609620419 = 874^3 + 06096^3 + 20419^3 (13 digits)
56482546234141 = 5648^3 + 25462^3 + 34141^3 (14 digits)
127684145437883 = 12768^3 + 41454^3 + 37883^3 (15 digits)

I found now the primes
455907121379 = 4559^3 + 7121^3 + 379^3 (12 digits) and
205143581217055139 = 205143^3 + 581217^3 + 055139^3 (18 digits)12775266502422183857 = 1277526^3  +  650242^3  +  2183857^3 (20 digits) and
20104465640022276577 = 2010446^3  +  564002^3  +  2276577^3 (20 digits)

The method I'm using is more of less the following.

We are looking for 3 numbers A,B,C and two suitable powers of 10, P and Q, such that  P*A + Q*B + C = A^3  +  B^3  +  C^3.
For example, 2213 = 2^3 + 2^3 + 13^3 corresponds to A=2, B=2, C=13, P=10^3 and Q=10^2.

I rewrite it as C^3 - C = P*A-A^3  +  Q*B - B^3, i.e.,
(C-1)*C*(C + 1) = P*A-A^3  +  Q*B - B^3.

So, I fix two specific values of P and Q (i.e. two specific
values for the number of digits in B and C) and
I compute V = P*A-A^3  +  Q*B - B^3 for all the values A and B such that
B does not exceed the number of digits prescribed by P and Q,
and  0 <= V < Q^3-Q.
Given a value of V it is then easy to see if there is a solution of
V = (C + 1)C(C-1), because if V=0 then C=0, or C=1, otherwise
it is sufficient to take the cubic root R of V and check if R^3-R is
equal to V again. The fact that C^3-C must be divisible by 6 allows avoiding some cases.

If one generalizes to more than 3 addends and to powers differents from
cubes, then there are several primes that can be expressed as sum of powers of their group of digits. For example:

101 = 10^2 + 1^2,
2213 = 2^3 + 2^3 + 13^3,
360091 = 3^2 + 600^2 + 9^2 + 1^2,
999371 = 999^2 + 37^2 + 1^2,
1004161 = 100^3 + 4^3 + 16^3 + 1^3,
1175791 = 1^3 + 1^3 + 75^3 + 7^3 + 91^3,
2230433 = 22^4 + 30^4 + 4^4 + 33^4,
4160407 = 4^3 + 160^3 + 40^3 + 7^3,
5814499 = 5^4 + 8^4 + 14^4 + 49^4 + 9^4,
5882353 = 588^2 + 2353^2,
8071249 = 8^5 + 0^5 + 7^5 + 1^5 + 24^5 + 9^5,
31480621 = 314^3 + 80^3 + 6^3 + 21^3,
79386649 = 7^5 + 9^5 + 38^5 + 6^5 + 6^5 + 4^5 + 9^5,
100118099 = 100^4 + 1^4 + 18^4 + 0^4 + 9^4 + 9^4,
174781559 = 1^3 + 7^3 + 47^3 + 8^3 + 1^3 + 559^3,
209385853 = 209^3 + 38^3 + 585^3 + 3^3,
333667001 = 333^3 + 667^3 + 0^3 + 0^3 + 1^3,
333667153 = 333^3 + 667^3 + 1^3 + 5^3 + 3^3,
562614211 = 56^5 + 26^5 + 1^5 + 4^5 + 2^5 + 1^5 + 1^5,
562633571 = 56^5 + 26^5 + 3^5 + 3^5 + 5^5 + 7^5 + 1^5,
562648243 = 56^5 + 26^5 + 4^5 + 8^5 + 2^5 + 4^5 + 3^5,
758898403 = 7^5 + 58^5 + 8^5 + 9^5 + 8^5 + 40^5 + 3^5,
2217388933 = 2^4 + 217^4 + 3^4 + 8^4 + 8^4 + 9^4 + 3^4 + 3^4,
3600000011 = 3^2 + 60000^2 + 0^2 + 0^2 + 1^2 + 1^2,
3615336233 = 3^3 + 6^3 + 1533^3 + 6^3 + 233^3,
5312388031 = 5^5 + 31^5 + 23^5 + 88^5 + 0^5 + 3^5 + 1^5,
9421127533 = 9^3 + 4^3 + 2112^3 + 75^3 + 33^3,
9764820899 = 9^5 + 7^5 + 6^5 + 48^5 + 2^5 + 0^5 + 8^5 + 99^5,
10000001371 = 100000^2 + 0^2 + 1^2 + 37^2 + 1^2,
10000092727 = 100^5 + 0^5 + 0^5 + 0^5 + 9^5 + 2^5 + 7^5 + 2^5 + 7^5,
10835826913 = 10^5 + 83^5 + 58^5 + 2^5 + 6^5 + 91^5 + 3^5,
10847047487 = 1^5 + 0^5 + 84^5 + 70^5 + 4^5 + 7^5 + 4^5 + 87^5,
11702270653 = 1^3 + 170^3 + 2270^3 + 65^3 + 3^3,
12596795599 = 1^5 + 25^5 + 9^5 + 6^5 + 79^5 + 5^5 + 5^5 + 99^5,
13609109243 = 13^5 + 60^5 + 91^5 + 0^5 + 92^5 + 4^5 + 3^5,
15625004509 = 15^3 + 6^3 + 2500^3 + 4^3 + 5^3 + 0^3 + 9^3,
17363292197 = 17^4 + 363^4 + 2^4 + 9^4 + 2^4 + 19^4 + 7^4,
18180110737 = 18^5 + 1^5 + 8^5 + 0^5 + 110^5 + 73^5 + 7^5,
20661274457 = 2^3 + 0^3 + 6^3 + 61^3 + 2744^3 + 5^3 + 7^3,
31179644707 = 3117^3 + 964^3 + 4^3 + 7^3 + 0^3 + 7^3,
31454703409 = 3145^3 + 4^3 + 703^3 + 4^3 + 0^3 + 9^3,
36331268291 = 36^3 + 3312^3 + 6^3 + 8^3 + 2^3 + 91^3,
38648433763 = 386^3 + 484^3 + 3376^3 + 3^3,
41100345113 = 4^3 + 1^3 + 100^3 + 3451^3 + 13^3,
46473009859 = 464^4 + 73^4 + 0^4 + 0^4 + 98^4 + 5^4 + 9^4
and so on

One may also ask if there are decompositions in which the single addends
are prime powers (not necessarily the resulting number).
In 2213 = 2^3 + 2^3 + 13^3,  2, 2 and 13 are all primes (and 2213 also).

The next such case (the only one I found so far) is
196731735813 = 19^3 + 673^3 + 17^3 + 3^3 + 5813^3, where
196731735813 is not prime, but 19, 673, 17, 3 and 5813 are all primes.

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