Three non-idiotic guesses

Three problems to ponder for the next 100 years:

• First one may seem contrived but thinking about this would hightlight some major properties of primes. So here it is : The number 82 is the largest positive number that cannot be written as sum of two square free numbers each having odd number of prime factors.
  Motivation for this guess comes from the result of the following python program that was run on sage to compute the number of such pairs of numbers summing upto a given number.

  def sqfreesum(n):
   sum = 0;
   for i in range(1,n ):
    sum +=moebius(i)^2* moebius(n-i)^2
      *( ( 1 + moebius(i))*( 1 + moebius(n-i) )/4 );
   return sum;

  
  The graph which shows the number of ways of writing a number as a sum of two such square free numbers, reveals a trend:it increasingly stays away from zero. In fact after taking off from 82, where it is zero, it never seems to drop down.

  Note that we can always write a number as a sum of two square free numbers. The numbers of ways of writing them is:
  . Have to warn you that I have not seen an actual proof but I have seen it mentioned here:
  A098235

• The second one is in the same spirit as the previous one. Any even number > 10 can be written as the sum of two square free numbers with one having odd number of prime factors and the other having even number of prime factors.
  Motivation for this guess comes from the the result of the following python program that was run on sage to compute the number of such pairs of numbers summing upto a given number.

  def sqfreesum(n):
   sum = 0;
   for i in range(1,n ):
    sum +=moebius(i)^2* moebius(n-i)^2
      *( ( 1 + moebius(i))*( 1 – moebius(n-i) )/4 + ( 1 – moebius(i))*( 1 + moebius(n-i) )/4);
   return sum;

  
  As with the previous conjecture the graph which shows the number of ways of writing a number as sum of such square free numbers. has only a few hiccups taking off, and at n=10 it achieves escape velocity. I should also mention here that any even number according to Chen Jigrun can be written as the sum of prime and semi-prime(product of two primes) or a sum of two primes. So when diluted, this result that says if Goldbach conjecture were false, we atleast know that every even positive number can be written as sum of two square free positive numbers one with odd number of prime factors( infact 1) and the other with even number of factors ( in fact 2).

• Treat the following with caution please. If denotes the Euler totient function, then the following number is always greater than 2n.
  .
  Proving this will prove the Goldbach conjecture. This is because, it is only when both i and 2n-i are prime that: is 1. There are better ways to reformulate the Golbach conjecture. One other way is the following:
  Given an even number $2n$, there exists a number such that