Problems & Puzzles:
JM Bergot sent
another nice puzzle:
I notice that 7+11=18 and
13+17+19+23=72. The pattern is for 3N consecutive primes:
the sum of the first N divides the sum of the second 2N.
Q1. Find a moderately large
example for N=2
How large N examples can you find?
Contributions came from Farideh Firoozbakht & Enoch Haga.
Answer to Q1:
I found two smaller solutions for N=2
3+5=8 ;7+11+13+17=48 and 5+7=12 ;11+13+17+19=60
But I haven't found larger solution for N=2.
Answer to Q2.
I found the following two solutions for N=4.
19+23+29+31=102 ; 37+41+43+47+53+59+61+67=408
47+53+59+61=220 ; 67+71+73+79+83+89+97+101=660
And for N=19 the solution is 37+41+...+113=1433 , 127+131+...+337=8598.
If the ratio of 1 to 2 primes is to be maintained no matter the
value of N,
if N=2 then 3,5,7 are the only starting primes producing an integral
solution for a total of 6 primes.
If the value of N=3, a total of 9 primes, maintaining the 1 to 2
a starting value of 2 produces an integral solution, namely 2+3+5=10
7+11+13+17+19+23=90, so that 90/10 = 9.
If the value of N=4, a total of 12 primes, maintaining the 1 to 2
find primes 19+23+29+31=102 and 37+41+43+47+53+59+61+67=408 so that
408/102=4. Also 47-61=220 and 67-101=660, so that 660/220=3.
Since the problem assumes the ratio is maintained, there will be an
number of solutions greatly outnumbered by those non-solutions