Problems & Puzzles: Puzzles

Puzzle 1159 Similar to Puzzle 48 Brian Trial sent the following nice puzzle, Similar to Puzzle 48 I was wondering if other powers of primes could be found that were the sum of powers of primes, with no primes repeated (all unlike). For example, if a = 5, the prime 1709 can be decomposed into the sum of 7 other 5th powers of unlike primes: 1709^5 = 1567^5 + 1373 ^ 5 + 719 ^ 5 + 503 ^ 5 + 431 ^ 5 + 367 ^ 5 + 349 ^ 5 Q1: Are there other P^5 that can be decomposed into the sum of 7 unlike Q^5? Q2: Is there P^5 that can be decomposed into less than 7 unlike Q^5? Q3: Decompose 661^7 into a sum of 7th powers of unlike prime numbers. Q4: Decompose 127 ^ 4 into a sum of 4th powers of unlike prime numbers.

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From Jan 20-26, 2024 contributions came from Michael Branicky, Emmanuel Vantieghem, Giorgos Kalogeropoulos, Gennady Gusev, Oscar Volpatti, Emmanuel Vantiegehem

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Michael wrote:

127^4 =  3^4 + 5^4 + 7^4 + 11^4 + 13^4 + 17^4 + 23^4 + 29^4 + 31^4 + 43^4 + 47^4 + 59^4 + 61^4 + 67^4 + 73^4 + 89^4 + 103^4
(a sum of 17 distinct primes)

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Emmanuel wrote:

there are three in the announcement of puzzle 1159 :

there are three places where we find  "*"  where we should find "^".

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Giorgos wrote:

Also, the expression given by Brial was first discovered by Michael Lau, 07/20/2002

you can visit   http://euler.free.fr/database.txt    for more records and details

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Gennady wrote:

Q1.

3 previously known solutions have been found again + 4 new ones for P<5100:

2843^5=2731^5+1933^5+1459^5+709^5+509^5+271^5+31^5 (Jean-Charles Meyrignac, 09/13/2000)

1709^5=1567^5+1373^5+719^5+503^5+431^5+367^5+349^5 (Michael Lau, 08/20/2002)

4027^5=3301^5+3169^5+3037^5+2411^5+1481^5+859^5+569^5 (Takao Nakamura, 2/10/2008)

4513^5=3673^5+3347^5+3313^5+3041^5+2473^5+1979^5+1087^5

4519^5=4409^5+2447^5+2293^5+2267^5+1319^5+1283^5+101^5

4861^5=4513^5+3329^5+3217^5+2423^5+1381^5+601^5+197^5

5023^5=4013^5+4007^5+3911^5+2767^5+1993^5+1693^5+1039^5

 

Q3.

661^7=607^7+521^7+503^7+421^7+419^7+383^7+317^7+227^7+211^7+197^7+151^7+97^7+71^7+41^7+23^7

661^7=563^7+557^7+487^7+479^7+463^7+443^7+293^7+281^7+239^7+193^7+83^7+47^7+31^7+17^7+13^7

 

Q4.

127^4=103^4+89^4+73^4+67^4+61^4+59^4+47^4+43^4+31^4+29^4+23^4+17^4+13^4+11^4+7^4+5^4+3^4

 

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Oscar wrote:

I found one more decomposition of a 5th prime power into the sum of 7 unlike 5th prime powers:

2843^5 = 2731^5 + 1933^5 + 1459^5 + 709^5 + 509^5 + 271^5 + 31^5.

 

Maybe no 5th prime power requires less than 7 powers, but I have no proof about it.

Searching among signed primes, I still found only decompositions with length at least 7, like:

101^5 = 97^5 + 83^5 + (-67)^5 + (-59)^5 + 31^5 + 29^5 + (-13)^5.

 

Prime power 661^7 can be decomposed into the sum of unlike 7th prime powers in many ways. My shortest solution requires 19 powers: 

661^7 = 653^7 + 409^7 + 401^7 + 313^7 + 311^7 + 283^7 + 263^7 + 251^7 + 227^7 + 197^7 + 179^7 + 157^7 + 137^7 + 103^7 + 89^7 + 83^7 + 79^7 + 43^7 + 11^7

There are several solutions with length 19 or more, but I don't know if 661^7 admits shorter solutions, as I didn't complete an exaustive search.
 

Searching among signed primes, 7th prime powers with shorter decompositions can be found quickly, like:

173^7 = 163^7 + 157^7 + (-139)^7 + 137^7 + (-127)^7 + (-113)^7 + 109^7 + (-97)^7 + (-79)^7 + 67^7 + 61^7 + 59^7 + (-37)^7 + 23^7 + (-11)^7.

 

Prime power 127^4 can be decomposed into the sum of 17 unlike 4th prime powers: 

127^4 = 103^4 + 89^4 + 73^4 + 67^4 + 61^4 + 59^4 + 47^4 + 43^4 + 31^4 + 29^4 + 23^4 + 17^4 + 13^4 + 11^4 + 7^4 + 5^4 + 3^4.

In this case, exaustive search ensures that 127^4  is the smallest decomposable 4th prime power and that its decomposition is unique.
 

Moreover, it can be proven that every 4th prime power requires at least 17 unlike 4th prime powers.

 

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Emmanuel wrote:

This is the only result I could achieve about puzzle 1159 :

127^4 = 3^4 + 5^4 + 7^4 + 11^4 + 13^4 + 17^4 + 23^4 + 29^4 + 31^4 + 

        + 43^4 + 47^4 + 59^4 + 61^4 + 67^4 + 73^4 + 89^4 + 103^4.

 

 

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