Problems & Puzzles:
Puzzle 116. A=B+C, A*B*C is a primorial...
Nomoto asks for numbers A, B & C such that the following four rules
= B + C
is a primorial
B, & C are, each, a product of n of distinct
has found only two examples:
= 2 + 3
= 231 + 874 [ 5*13*17 =
3*7*11 + 2*19*23 ]
1: Find the next example
the condition 3 is relaxed and we let that A, B & C are the product of
distinct primes, not necessarily the same quantity, example:
= 3 + 7 [ 2*5 = 3
+ 7 ]
some more examples, being the largest currently found, the following one:
17x23x31x41 = 2x3x7x11x29x37 + 5x13x19 ]
2. Find the next 3 examples
3. Can you argue if this sequence is finite or infinite?
All the numbers for the sequence described in the question 2 are a subset
Hakan Summakoğlu wrote (May
Q2: 219965 = 213486 +
6479 [ 5x29x37x41 = 2x3x7x13x17x23 + 11x19x31]
Jan van Delden wrote (May 2011)
I was wondering about
the number of solutions of Puzzle 116, I thought that could be improved.
Must have missed this one..
We have P:=p[n]#=a.b.c and a=b+c. Given P there are two equations with 3
unknowns. We can therefore (for instance) choose b and a and c follow. From
this one can deduce that if b^2+4P/b is a square, say D^2, we have: a=(D+b)/2
Furthermore the minimum number of factors of P to be assigned to a can be
calculated beforehand (using the largest prime factors of P), leading to the
maximum number of factors to be assigned to b. Unfortunately this maximum is
about n/3, which still gives a lot of tests to perform!
I found the following 20 solutions, in order of appearance (primarily Q2):
35= 2+33 5x7=2+3x11
22= 7+15 2x11=7+3x5
1105=874+231 5x13x17=2x19x23+3x7x11 (Q1)
No further solutions with n<=35
I computed the minimum absolute distance between b^2+4P/b and the nearest
square (where no solution was found), let's call this dist(D^2). The
smallest distance 103 was found (n in [12,14-26]) at n=18, b=12509182. At
first sight this is large, however a and c fail to be integer by a mere
0.000000133 (=dist(a),dist(b)). However the resulting distance between P and
a.b.c is huge, since a.b.c=(D+b)/2.b.(D-b)/2=b.(D^2-b^2)/4, this gap
is of the order b.dist(D^2)/4=12509182.103/4.
In general dist(a)=dist(b)=dist(D^2)/(4D) and dist(P)=b.dist(D^2)/4. And
dist(D^2) is maximal D+1/4. More should be known about the distribution of
dist(D^2) to say more.