Problems & Puzzles: Puzzles

Puzzle 877. Integers represented using the minimal first consecutive primes...

Abhiram R. Devesh sent the following nice puzzle.

I have another puzzle for your website. This is inspired from the Numberphile video ( https://youtu.be/pasyRUj7UwM) and in turn by the paper by Mr. Taneja (https://arxiv.org/abs/1302.1479). The puzzle is as follows:

Can we represent natural numbers in terms of prime numbers in increasing and decreasing order?

The conditions are as follows
a. There is no limit to the list of prime numbers used.

b. But the target is to use minimum number of primes as possible.

c. The increasing order of primes must begin with 2 and the decreasing order of primes must end with 2.

d. We can use the following operators and auxiliary symbols
• Product & Division
• Power
• Concatenation
• Brackets
Here are some examples sent by Abhiram (and a few others added by Carlos Rivera just off hand...)

 Number, N Increasing order Author Decreasing order Author 1 2*3-5 CR 3-2 ARD 2 2 ARD 2 ARD 3 2^3-5 CR 7-5+3-2 ARD 4 2^(-3+5) CR (5+3)/2 CR 5 2+3 ARD 3+2 ARD 6 2*3 ARD 3*2 ARD 7 2+3-5+7 ARD 7-5+3+2 ARD 8 2^3 ARD 9 3^2 ARD 10 2+3+5 ARD 5+3+2 ARD 11 (2*3)+5 ARD 5+(3*2) ARD 12 23-(5+7) ARD 13 2^3+5 CR 5*3-2 CR 14 5+3^2 CR 15 16 (5+3)*2 CR 17 2+3*5 CR 5*3+2 CR 18 23-5 ARD 7+5+3*2 ARD 19 20 21 23+5-7 ARD ARD 22 23 23 ARD ARD

Q. Please send your best results for N<=100. Please use the same format as the Table above.

Contributions came from Claudio Meller, Dmitry Kamenetsky, Michael Hürt and Carlos Rivera

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Claudio, Dmitry & Michael results are combined in the following table. In red are these results that used larger primes than the other contributions.

 Increasing order Decreasing order N ARD&CR CM MH DK ARD&CR CM MH DK 1 2*3-5 -2+3 (2+3)/5 3-2 3-2 3-2 2 2 2 2 2 2 2 3 2^3-5 2^3-5 2^3-5 7-5+3-2 7-5+3-2 7-5+3-2 4 2^(-3+5) 2-3+5 2-3+5 (5+3)/2 -5 + 3^2 = (5+3)/2 (5+3)/2 5 2+3 2+3 2+3 3+2 3+2 3+2 6 2*3 2 x 3 2*3 3*2 3x2 3*2 7 2+3-5+7 2+3-5+7 2+3-5+7 7-5+3+2 7 + 5 - 3 - 2 7+5-3-2 8 2^3 2^3 2^3 ((7+5)/3 ) x 2 (7+5)/3*2 (7-5)+(3*2) 9 2x(3+5)-7 2*(3+5)-7 (-2+3)*(5-7+11) 3^2 3^2 3^2 10 2+3+5 2+3+5 2+3+5 5+3+2 5+3+2 5+3+2 11 (2*3)+5 2x3 + 5 2*3+5 5+(3*2) 5+3x2 5+3*2 12 23-(5+7) Wrong 2 x 3 x (-5+7) ((2+3)/5)^7+11 7 + 5 x (3 -2) (7-5)*3*2 (7-5)*3*2 13 2^3+5 2^3 + 5 2^3+5 5*3-2 5 x 3 - 2 5*3-2 14 23 - 5 + 7 -11 2+3+5-7+11 -(2*3)+5*(-7+11) 5+3^2 5 + 3^2 5+3^2 15 2x3 + 5 -7 + 11 2*3+5-7+11 -2-3+5*(-7+11) (11-7) x 5 - 3 -2 75/(3+2) (11-7)*5-3-2 16 2 x (3+5) 2*(3+5) 2*(3+5) (5+3)*2 (5+3) x 2 (5+3)*2 17 2+3*5 2 + (3x5) 2+3*5 5*3+2 5 x 3 + 2 5*3+2 18 23-5 23 - 5 23-5 7+5+3*2 -7 + 5 x (3+2) (7+5-3)*2 19 2 x 3 - 5 + 7 + 11 (2+3)/5+7+11 2+35-7-11 (7 x 5 +3) /2 (7*5+3)/2 (11-7)*5-3+2 20 -2 + 3x5 +7 2^3+5+7 -2+(3*5)+7 7 + 5 x 3 - 2 7+5*3-2 7+(5*3)-2 21 23+5-7 23 + 5 -7 23+5-7 (11-7) x 5 + 3 - 2 7+5+3^2 (11-7)*5+3-2 22 2^3 x 5 - 7 - 11 2-3+5+7+11 23-5-7+11 (11 x (7+5)/3 ) / 2 11+7+5-3+2 11+7+(5+3)/2 23 23 23 23 7 + (5+3) x 2 75/3-2 7+(5+3)*2 24 2 + 3x5 +7 2+3*5+7 7 + 5x3 + 2 7+5*3+2 25 (2+3) x 5 (2+3)*5 5 x (3+2) 5*(3+2) 26 -2 + 35 - 7 2*3*5+7-11 -11 + 7 + 5 x 3 x 2 7*5-3^2 27 -2^3 + 5 x 7 2/3*57-11 -5 + 32 75/3+2 28 23 + 5 23+5 7 x ( 5+3) /2 7*(5+3)/2 29 -2 x 3 + 5 x 7 (2+3)*5-7+11 7 x 5 -3 x 2 7*5-3*2 30 2 x 3 x 5 2*3*5 5 x 3 x 2 5*3*2 31 -2 + 3 x 5 + 7 + 11 2^3+5+7+11 11 + 7 + 5x3 - 2 11-7-5+32 32 2^(35/7) (2+3)*5+7 32 32 33 -2 + 35 2^3*5-7 -11 - 7 + 53 -2 11*(7-5)*3/2 34 -23 + 57 2-3+5*7 7 - 5 + 32 (7+5)*3-2 35 23 + 5 + 7 23+5+7 7 x 5 x (3-2) 7*5*(3-2) 36 -2 + 3 + 5 x 7 23-5+7+11 (75-3) / 2 ((7-5)*3)^2 37 2 + 35 2+35 5 + 32 5+32 38 23 x 5 - (7 x 11) 2/3*57 -7 + 5 x 3^2 (7+5)*3+2 39 2 x 3 + 5 x 7 +11 -13 (2-3+5)*7+11 (75+3)/2 (75+3)/2 40 2^3 x 5 2^3*5 7x5 + 3 + 2 7*5+3+2 41 2 x 3 + 5 x 7 2*3+5*7 7x5 + 3x2 7*5+3*2 42 (-2+3+5) x 7 (2+3)*5-7+11+13 7 x (5+3-2) 7*(5+3-2) 43 2^3 + 5 x 7 2^3+5*7 75 - 32 75-32 44 2 +35 +7 2+35+7 -7 +53 -2 7+5+32 45 -23 +57 +11 2-3+5*7+11 5 x 3^2 5*3^2 46 23 x (-5+7) 23+5+7+11 11 + 7x5 x (3-2) ((11-7)*5+3)*2 47 2^3 x 5 + 7 2^3*5+7 -11 + 7 + 53 -2 7^(5-3)-2 48 (2+ 3x5 +7) x (-11+13) 2*3*5+7+11 -7 + 53 +2 (11*7-5)/3*2 49 -2^3 +57 (2+3)*(5+7)-11 11 -7 + 5 x 3^2 11-7+5*3^2 50 (23-5+7) x (-11+13) 2*(3*(5+7)-11) (75/3) x 2 75/3*2 51 -2 x 3 + 57 2+3+5*7+11 53-2 53-2 52 -2 - 3 + 57 2*3+5*7+11 7 + 5 x 3^2 7+5*3^2 53 (-2+3+5) x  7 + 11 ((2-3)^5+7)*11-13 11 + 7 x (5+3-2) 11*(7-5+3)-2 54 -2 + (3+5) x 7 (2^3-5)*(7+11) 7 x (5+3) - 2 7*(5+3)-2 55 2 +35 + 7 + 11 2+35+7+11 53+2 53+2 56 2 - 3  + 57 2-3+57 -11 + 7x5 +32 (11-7)*(5+3^2) 57 (-2 + 3) x 57 2+(3-5+7)*11 -7 + (5+3)^2 11*(7-5+3)+2 58 23 + 5 x 7 = -2 +3 +57 23+5*7 7 + 53 -2 7+53-2 59 (2+3+5) x 7 -11 (2+3+5)*7-11 11 - 7 + 53 +2 11-7+53+2 60 (2+3) x (5+7) (2+3)*(5+7) (7 + 5) x (3 +2) (7+5)*(3+2) 61 (2x3) x (5+7) -11 2*3*(5+7)-11 -11+ (7+5)x3 x2 11*7-(5+3)*2 62 2 + 3 + 57 2+3+57 7+53+2 7+53+2 63 2 x 35 - 7 =  2 x3 + 57 2*3+57 -11 +75 -3 +2 11*7-5-3^2 64 (2^3)^ (-5+7) 2-3*5+7*11 (5+3)^2 (5+3)^2 65 2^3 +57 2^3+57 11 + 7 x (5+3) -2 (11+7-5)*(3+2) 66 -(2x3+5) + 7x11 ((2-3)^5+7)*11 75 - 3^2 75-3^2 67 2^3 + 5x7 + 11 +13 2-3+57+11 7 x 5 + 32 7*5+32 68 2 x 35 + 7 -11 2-3*(5-7)*11 11 - 7 + (5+3)^2 11-7+(5+3)^2 69 23 +  57 - 11 23+5*7+11 75 - (3x2) 75-3*2 70 2 x 35 2*35 75 - 3 -2 75-3-2 71 (2+3) x (5+7) + 11 2-3-5+7*11 7 + (5+3)^2 7+(5+3)^2 72 2x3 x (5+7) 2*3*(5+7) (7+5) x 3 x 2 (7+5)*3*2 73 2 + 3 + 57 + 11 2+3+57+11 11 +7 + 53 +2 11*7-5+3-2 74 2 x 35 - 7 + 11 2*3+57+11 75 - 3 + 2 75-3+2 75 ((2^3)^(-5+7)) +11 ((2+3)/5+7)*11-13 75 x (3-2) 75*(3-2) 76 2^3 + 57 + 11 2*(3+5*7) 75 + 3 -2 (7*5+3)*2 77 2 x 35 + 7 (2*3+5)*7 7 x (5+3x2) 7*(5+3*2) 78 (2x3-5) + 7 x 11 (2+3)/5+7*11 11 + 7x5 +32 (11+7-5)*3*2 79 2 x 35 + 7 -11 + 13 ((2-3)^5+7)*11+13 -13 + 11 +75 +(3x2) ((13+11+7)*5+3)/2 80 23 + 57 23+57 75 + 3 + 2 75+3+2 81 2 - 3 + 5 + 7x11 (2+3+5)*7+11 75 + (3x2) (7+5-3)^2 82 (2x3 + 5x7 ) x (-11+13) 23-5+7*11-13 11 + 7  + (5+3)^2 11*7+5*(3-2) 83 2x3x(5+7) + 11 2*3*(5+7)+11 11 + (7+5) x 3 x 2 11*7+5+3-2 84 2 x (35+7) 2*(35+7) 75 + 3^2 75+3^2 85 2^3 x (5+7) -11 2^3*(5+7)-11 11 +75 - 3 + 2 (11+7)*5-3-2 86 2 x (35+7) -11 + 13 2+3+57+11+13 11 + 75  x (3-2) (11+7*5-3)*2 87 2+3+5+ (7x11) 2+3+5+7*11 11 + 75 + 3 - 2 11*7+5+3+2 88 2 x 35 + 7 + 11 ((2+3)/5+7)*11 11 + 7 x (5+3x2) 11*7+5+3*2 89 2^3 + 57 + 11 + 13 2-3+5*(7+11) -17 - 13 - 11 + 7 +5^3-2 (11+7)*5-3+2 90 2^3 + 5 + 7 x 11 2^3+5+7*11 117 + 5 -32 (11+7)*5*(3-2) 91 (-2+3x5) x 7 (2^3+5)*7 7 x (5x3-2) 7*(5*3-2) 92 -2 + 3x5x7 - 11 23+5+7*11-13 11 + 75 + 3 x 2 11+75+3*2 93 2x (3+5) + 7x11 2*(3+5)+7*11 11x7 + (5+3)x2 11*7+(5+3)*2 94 2 +  3x5 + 7x11 2+(3*5)+7*11 11x7 + 5x3 +2 11*7+5*3+2 95 2 x (35 + 7) + 11 23-5+7*11 11 +75 + 3^2 (11+7)*5+3+2 96 2^3 x (5+7) 2^3*(5+7) -11 +75 + 32 (11+7)*5+3*2 97 23 x 5  -7 - 11 23*5-7-11 -17 - 13 +11 - 7 + 5^3 - 2 11*(7+5-3)-2 98 2^3 x  (5+7) - 11 + 13 2^3+5*(7+11) 7 x ( 5 + 3^2) 7^(5-3)*2 99 23 + 57 - 11 + 13 +17 (2*(3+5)-7)*11 -7 + 53 x 2 (11+7)*5+3^2 100 (2+3) x 5 x (-7+11) 2+3+5+7*11+13 (11 + 7 -5 -3)^2 (11+7-5-3)^2

***

Carlos Rivera just makes the following comment:

According to the condition imposed to this puzzle (a, c & d) by Abhiram R. Devesh, ALL integers N can be represented in several ways as an algebraic sum of primes, that is to say, using only the operators + & -, without using the operators *, / and ^. In this case these solutions could use not the minimal quantity possible of primes.

In other words there is not a possible "hole" in this table.

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