Problems & Puzzles: Puzzles

Puzzle 646. Curio 364 I will stole another puzzle to my friend Claudio Meller. In one of his pages we may appreciate this interesting curio: 3642 = 132 496 & 496 − 132 = 364 3643 = 48 228 544 & 544 − 228 + 48 = 364 3644 = 17 555 190 016 & 016 − 190 + 555 − 17 = 364 3645 = 6 390 089 165 824 & 824 − 165 + 089 − 390 + 6 = 364 3646 = 2 325 992 456 359 936 & 936 − 359 + 456 − 992 + 325 − 2 = 364 3647 = 846 661 254 115 016 704 & 704− 016 + 115 − 254 + 661 − 846= 364 Q. Would you like to try a prime as/or better than 364?

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Contributions came from Ryan Bailey, Hakan Summakoğlu

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Ryan wrote:

As far as the specific situation you described no prime solutions exist. Although 78 goes up to 4, 287 goes up to 4, and as shown 364 goes up to 7.

If each substring is allowed to be of length 4, 1096 goes up to 3.

If each substring is allowed to be of length 5, 18183 goes up to 4.

If each substring is allowed to be of length 6, 336634 goes up to 3.

If each substring is allowed to be of length 7, 2727274 goes up to 4.

And I found nothing for a length of 8.

Sadly none of the solutions above are prime, nor do they go higher than 7.

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Hakan wrote:

I could not find prime result for <5.10^9. My best result is 7th power too, for 288553552.
 

288553552² = 83263152 371816704 & 371816704-83263152 = 288553552

288553552³ = 24025878 367604934 632132608 & 632132608-367604934+24025878 = 288553552

288553552⁴ = 6932752 542892365 620829677 373423616 & 373423616-620829677+542892365-6932752 =288553552

288553552⁵ = 2000470 371388624 453573088 593115414 797484032 & 797484032-593115414+453573088-371388624+2000470 = 288553552

288553552⁶ = 577242 831334946 758472573 805154135 685767378 096881664 & 096881664-685767378+805154135-758472573+831334946-577242 = 288553552

288553552⁷ = 166565 469348235 788888147 266059381 759618132 795582204 270870528 & 270870528-795582204+759618132-266059381+788888147-469348235+166565 = 288553552

Good work Hakan!

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So, a new question arises: why NO prime solutions exist? Is this casual or not?

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Giovanni Resta wrote on July 31:

I searched up to 900*10^9 and there is only
a value which is better (or equal) to 288553552,
that is N=333366663334, for which the scheme can
be extended to 8-th power.
Here
N^2 = -111133332222+444499995556,
N^3 = +37048148148-185229628518+481548143704
and so on, always alternating signs, up to
N^8= -152537753+393690626684-973632191219
     +393376865929-117175314305+813083639737
     -452197172235+276372746496.

This value of N (333366663334) suggests that playing
a little with patterns it is possible to obtain
larger solutions. Indeed, a number with the same
pattern of N but with 1890 digits can be expressed
with powers up to the 14-th one.

But we can do much better, using numbers of
the form  5000050000 or 4999950000.

Indeed, it should be not too difficult to prove,
(using for example the expansion of the binomial) that
a number of the form above of 2*D digits can be
expressed with powers up to Ceil(D/2)*2, using only
the plus sign between the parts, as in

5000050000^6=
156259375+2343781250+2343759375+156250000+0+0

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