Problems & Puzzles: Puzzles

 Puzzle 863. 2017 a new prime year Emmanuel Vantieghem sent the following new year greetings: 2017 = 2*3*5 - 7 + 11*13*17 - 19*23, a nice arithmetic curio using ordered the first 9 primes. I have obtained this other curio, 2017 =2*7*11*13+3*5, using disordered the first 6 primes. Q1. Send your own nice curio for 2017 using only prime numbers. Digging around I found the following interesting other properties for 2017: 2017=9^2+44^2 (Fermat's 4n+1 theorem) 2017 is a pancake number, because a pancake can be divided into 2017 parts by 63 straight cuts. 2017=(63^2+63+2)/2.... (Numbers Aplenty, Giovanni Resta) 2017=1^3+3^3+4^3+5^3+6^3+7^3+8^3+9^3 (Claudio Meller) 2017 = 1331 + 686 (Claudio Meller) Q2. Send more interesting arithmetic properties for 2017.

Contributions came from Carlos Rivera, Giovanni Resta, Inder J. Taneja and Emmanuel Vantieghem.

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Rivera wrote:

For Q1. 2017=sum algebraic of all the primes from 2 to 139, being negative only 2 & 53.

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Giovanni wrote:

For Q2. 2017^(1/3)=12.634807593300104456522514362496, First 10 terms pandigital (Giovanni Resta)

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Inder published a paper where he calculated:

2017 = 12^3 + 4 × 56 + 7 × 8 + 9
2017 = 98 + 7 × 6 + 5^4 × 3 + 2 × 1

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Emmanuel wrote:

Here are a few presentations for the prime 2017 :

Q1.
2017 = 31-37+43+53+41*47 (consecutive, not ordered)
2017 = 3^(3^2) + (2*22)^2  (uses only the primes  2  and  3)

Q2.
1*2 + 3 + 4*(5 - 6 + 7*8*9) = 2017 = (9*8*7 - 6 + 5)*4 + (3 + 2)*1
2017 = 5 + 6 + 7*8*9 + 10*11*12 + 13*14
2017 = -7 + 8 + 9*10 + 11*12*13 + 14*15
2017 = 21^2 + 22^2 - 23^2 - 24^2 + 25^2 + 26^2 - 27^2 + 28^2 + 29^2
2017 = 4^2 * 5^2 + (6^2)*(7^2) + 8^2 + 9^2 - 10^2 + 11^2 - 12^2 - 13^2
2017 = 10^3 +9^3 + 8^3 - 7^3 + 6^3 - 5^3 +4^3 - 3^3 -2^3 - 1^3
2017 = 2^11 - 2^5 + 1
2017 = 7^0 + 7^1 - 7^2 - 7^3 +7^4

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Jean Brette wrote on Jan 18, 2017:

2017 is a full reptend prime, i.e.  a prime number  p  such that the length of the period of  1/ p   is maximal and equal to (p-1). In this case , the period is a cyclic number. The smallest reptend prime is 7. The period of 1/7  is 142857.

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José de Jesús Camacho wrote on Jan 19, 2017:

2017 = 2^(2+3^2) - (2^2+3^3)

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On Jan 14, 2019, Pierandrea Formusa wrote:

For Q1

We can express 2017 like the sum of a prime and the square of a prime and the cube of a prime in ten number of ways, with all primes different:

 p1 p2 p3 p1 p2^2 p3^3 sum 1723 13 5 1723 169 125 2017 1531 19 5 1531 361 125 2017 1051 29 5 1051 841 125 2017 523 37 5 523 1369 125 2017 211 41 5 211 1681 125 2017 1553 11 7 1553 121 343 2017 677 3 11 677 9 1331 2017 661 5 11 661 25 1331 2017 397 17 11 397 289 1331 2017 157 23 11 157 529 1331 2017

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