Problems & Puzzles: Puzzles

Puzzle 32.- Find couples of numbers like this (1033, 8) such that:
1033 = 8^1+8^0+8^3+8^3

(Proposed by Patrick De Geest, 9/12/98)

This means to find couples of numbers (N, B) such that N = sum( B^di ); di’s are the decimal digits of N. Some other examples found by Patrick are the following:

 N B Comments about N 1 1 Trivial 10 9 12 3 100 98 101 50 Palprime 111 37 Palindrome 1033 8 Prime 2112 32 Palindrome 4624 4 20102 100 Palindrome 31301 25 595968 4 1053202 16 3909511 5 13177388 7 52135640 19

Can you find other prime and/or palindrome numbers N and its corresponding base B ?

Solution

Felice Russo (4/06/99) sent solutions (N,B), several for N=prime, one for N=Palprime, and two more for N=Palindrome:

(N prime, B) =
(101111, 20222)
(1010203,100)
(11111101,1587300)
(101001001, 25250249)

(N palprime, B) =
(111010111,15858587)

(N palindrome,B) =
(111101101101111, 9258425091759)
(101010000010101, 16835000001682)

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This is the Patrick De Geest's comment about the search of Russo:

"...I didn't noticed it that with restricting oneself to binary digits '0' and '1' one could construct more solutions. Nice, especially the new palprime 111010111 with base 15858587. Searching for cases of N * X + d. (111010111 = 15858587 x 7 + 2 --> 7 units and 2 zeros)...Felice's approach clearly is more effective. Congratulations..."

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According to the Patrick's comment, thanks to Russo's work, we have now at hand a method to find larger ans larger examples of this kind, eh...?

***

The 13/8/2001 Tiziano Mosconi found the least case (N,B) where N is palprime and B is prime:

{11100000011111000000111, 1009090910101000000009}

After this he found other 6 cases.

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Anurag Sahay sent the following table (Jan 2005):

 N B Comments about N 1224103 33 102531321 40 2758053616 15 7413245658 17 1347536041 20 2524232305 66 4303553602 40 16250130152 50 25236435456 48 35751665247 29 405287637330 28 419370838921 18 PRIME 835713473952 21 985992657240 19 1035263675371 47 PRIME 5063106413637 59 3606012949057 23 5398293152472 24 6821803782221 35 7560550222541 69 PRIME 67850843167550 49 60383799081932 30 15554976231978 27 27348457391264 31 14375256503038 44 203200576992435 36 162455874783801 52 407374116975631 42 662953283249375 41 1382570662371937 48 3427807121982740 53 17918292835887535 59 27303134872679943 62 36609478778945537 64 PRIME 42075899760411857 65

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Later, on September 2005,  Anurag Sahay sent more examples:

(6035143058499257267, 113)
(680167723454978641921, 191)
(1470730546922115245998, 199)
(2459025789447836964220840587, 978)
(5144388812832249823657004916963778, 4928)

(36083215428634110977759 PRIME, 297)
(4034921884375455037142415526879 PRIME,2329)

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On Set. 20, 06 Fred Schneider wrote:

I revisited this puzzle: I found one additional solution:  N=2210210, B=858  and found that one of the posted solutions was incorrect: N=5063106413637, B=59 is not a solution because the "power-digit-sum" = 2615920743659, not N itself.  All the others check out.

P.S. I'm very curious about Mr. Sahay's algorithm for finding these solutions.  It's very impressive

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Qu Shun Liang wrote (Dec 23, 2008) many solutions. I will show only the prime solutions:

419370838921 18 prime!
36609478778945537 64 prime!
139251164815885201 80 prime!
3302927389750318219 101 prime!
947288765894401687681 198 prime!
181959980982018167249 149 prime!
1738302526177719712897 212 prime!
8229212226515738544409 252 prime!
8261738498910420005081 252 prime!
3083035169354427026797 226 prime!
72374194476756069205463 321 prime!
36083215428634110977759 297 prime!
95819846877371720464703 331 prime!
72486576766263162199063 321 prime!
22597217327585752660759 282 prime!
116357702912903726409230779 675 prime!
289308896713546025759668637 771 prime!89423567665158371901674138556551281 7080 prime!

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