Problems & Puzzles: Puzzles

Puzzle 817. 2016  and prime numbers

I have found some curio expressions for 2016 in the web.

Three very interesting have been published by Claudio Meller in one of his pages. Here are them:

2016 = 2^5 + 2^6 + 2^7 + 2^8 +2^9 + 2^10

2016 = 666 + 666 + 666 + (6 + 6 + 6)

                (aaaa-aa)*aa
                 __________ - a - a

                       a+a

2016 =  _________________     valid if a is any of {0,1,2, ...,9}

                    a + a + a

By my own I have found two more:

2016 = T(63) = 63*64/2

2016 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3

But I have not find any interesting expression using only prime numbers, except the very-ugly/non-interesting following expression:

2016 = 71 + 73 + ... +151 +157 (18 consecutive prime numbers)

Q Send your best interesting expressions using only prime numbers?


Contributions came from Seiji Tomita

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Seiji wrote:

About Puzzle 817.

2016=a^3+b^3+c^3+d^3
a,b,c,d are primes.
{a,b,c,d}<5000
There are five solutions.

2016 = 349^3+ 277^3+ 167^3 -409^3
     = 463^3+ 419^3 -557^3 -13^3
     = 1123^3+ 571^3+ 149^3 -1171^3
     = 2153^3+ 1061^3+ 271^3 -2237^3
     = 3709^3+ 2099^3 -3877^3 -1259^3

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T. D. Noe wrote on Jan 24, 2016

There are more solutions to this puzzle in the form of one positive cube
and three negative ones.  For example, in the range up to 5000, there are
three additional solutions:

2016 = 1637^3 - 1399^3 - 1181^3 - 113^3

2016 = 1877^3 - 1867^3 - 443^3 - 263^3

2016 = 3457^3 - 3359^3 - 1423^3 - 811^3

There appear to be an infinite number of solutions.  I computed 72
solutions less than 200000. They are at
http://IntegerSequences.org/s000832.html

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