1 is a product of 0 primes such that every 2 positive integers that multiply to it add up to a prime.
1*1=1, 1+1=2
2 is a product of 1 prime such that every 2 positive integers that multiply to it add up to a prime.
1*2=2, 1+2=3
6=2*3 is a product of 2 primes such that every 2 positive integers that multiply to it add up to a prime.
1*6=6, 1+6=7
2*3=6, 2+3=5
30=2*3*5 is a product of 3 primes such that every 2 positive integers that multiply to it add up to a prime.
1*30=30, 1+30=31
2*15=30, 2+15=17
3*10=30, 3+10=13
5*6=30, 5+6=11
1. What is the smallest product of 4 primes such that every 2 positive integers that multiply to it add up to a prime?
2. What is the smallest product of 5 primes such that every 2 positive integers that multiply to it add up to a prime?
The first with 4 prime factors is 210, and the first with 5 prime factors is 186162.
The prime factors of 210 are
2 3 5 7
Its divisors are
1 2 3 5 6 7 10 14 and their complements in reverse order
15 21 30 35 42 70 105 210
All produce a prime when added to their complement, e.g., 10+21 = 31.
The prime factors of 186162 are
2 3 19 23 71
and its divisors are
1 2 3 6 19 23 38 46 57 69 71 114 138 142 213 426 and their complements
437 874 1311 1349 1633 2622 2698 3266 4047 4899 8094 9798 31027 62054 93081 186162
All produce a prime when added to their complement, e.g., 38+4899 = 38+4899 = 4937
% clearvars,clc
% p=primes(100);
% np=length(p);
% for a=1:np-5
% disp(a)
% for b=a+1:np-4
% for c=b+1:np-3
% for d=c+1:np-2
% for e=d+1:np-1
% i=p(a)*p(b)*p(c)*p(d)*p(e);
for i=1:186162
f=factor(i);
if length(f)>3
if isequal(f,unique(f))
dv=divisors(i);
good=true; lp=length(dv)+1;
for j=1:length(dv)/2
if isprime(dv(j)+dv(lp-j))==false
good=false;
break
end
end
if good
disp([length(factor(i)) i])
end
end
end
end
% end
% end
% end
% end
% end
The choice of 186162 for the ending of the search was based on the outer (now commented out) loop finding that worked, but it had time to check only even numbers before starting out with 3 instead of 2. The one-by-one check was to rule out lower numbers which happened to be odd, something that apparently does not happen.
Showing number of prime factors, and the number itself
>> primeSums
4 210
4 330
4 462
4 1870
4 4218
4 5590
4 6042
4 7638
4 13962
4 14410
4 22570
4 24682
4 26202
4 32838
4 35098
4 39678
4 43498
4 43990
4 52782
4 62458
4 62538
4 93262
4 125302
4 127290
4 130782
4 152310
4 155170
5 186162
The same without the test that the prime factors must be unique:
>> primeSums
4 210
4 330
4 462
4 1870
4 4218
4 5590
4 6042
4 7638
4 13962
4 14410
4 22570
4 24682
4 26202
4 32838
4 35098
4 39678
4 43498
4 43990
4 52782
4 62458
4 62538
4 93262
4 125302
4 127290
4 130782
4 152310
4 155170
5 186162
This is the same list. It seems that after 4, there are no other numbers with duplicate prime factors that satisfy the requirements.
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Posted by Charlie
on 2023-04-19 09:01:58 |
computer solution
