Coefficients of polynomials f_k(x)
The summing problem:
 
               
is converted by the well knwon Bernoulli-formula into a polynomial problem. Here the first parenthese on the rhs is converted into a bernoulli-polynomial in x and the second parenthese in a binomial-expression acting on the same x. If their difference is zero at a positive integer x then a solution for the problem is found. Thus the original problem is restated as a problem of positive integer zeros of a polynomial in x
Definition of the new polynomials:
                 
 
     
                 
Conjecture / to prove:
*    no positive integer real roots x do exist for any polynomial f_k(x) k>2    
                       
  list of unscaled polynomials f_k(x)              
K=
0 f0(x)= -1
1 f1(x)= -1         +1 x
2 f2(x)= -1      -1/2 x       +1/2 x^2
3 f3(x)= -1     -11/6 x      -1/2 x^2      +1/3 x^3
4 f4(x)= -1        -3 x     -11/4 x^2     -1/2 x^3       +1/4 x^4
5 f5(x)= -1   -121/30 x        -6 x^2    -11/3 x^3      -1/2 x^4       +1/5 x^5
6 f6(x)= -1        -5 x   -121/12 x^2      -10 x^3    -55/12 x^4      -1/2 x^5      +1/6 x^6
7 f7(x)= -1   -251/42 x       -15 x^2   -121/6 x^3       -15 x^4     -11/2 x^5     -1/2 x^6     +1/7 x^7
8 f8(x)= -1        -7 x   -251/12 x^2      -35 x^3   -847/24 x^4       -21 x^5   -77/12 x^6    -1/2 x^7    +1/8 x^8
... ... ...
                       
  list of factorial row-scaled polynomials F_k(x), coefficients are definitely integer      
K=
0 F0(x)= -1
1 F1(x)= -1         +1 x
2 F2(x)= -2         -1 x          +1 x^2
3 F3(x)= -6        -11 x          -3 x^2          +2 x^3
4 F4(x)= -24        -72 x         -66 x^2         -12 x^3          +6 x^4
5 F5(x)= -120       -484 x        -720 x^2        -440 x^3         -60 x^4         +24 x^5
6 F6(x)= -720      -3600 x       -7260 x^2       -7200 x^3       -3300 x^4        -360 x^5        +120 x^6
7 F7(x)= -5040     -30120 x      -75600 x^2     -101640 x^3      -75600 x^4      -27720 x^5       -2520 x^6       +720 x^7
8 F8(x)= -40320    -282240 x     -843360 x^2    -1411200 x^3    -1422960 x^4     -846720 x^5     -258720 x^6     -20160 x^7     +5040 x^8
... ... ...
                       
  list of monic polynomials M_k(x), (f_k(x) rowscaled to have monic polynomials)      
K=
0 M0(x)= -1
1 M1(x)= -1         + x
2 M2(x)= -2        -1 x        + x^2
3 M3(x)= -3     -11/2 x     -3/2 x^2        + x^3
4 M4(x)= -4       -12 x      -11 x^2       -2 x^3        + x^4
5 M5(x)= -5    -121/6 x      -30 x^2    -55/3 x^3     -5/2 x^4         + x^5
6 M6(x)= -6       -30 x   -121/2 x^2      -60 x^3    -55/2 x^4        -3 x^5       + x^6
7 M7(x)= -7    -251/6 x     -105 x^2   -847/6 x^3     -105 x^4     -77/2 x^5     -7/2 x^6     + x^7
8 M8(x)= -8       -56 x   -502/3 x^2     -280 x^3   -847/3 x^4      -168 x^5   -154/3 x^6    -4 x^7     + x^8
... ... ...
                       
  polynomials f_k(x) evaluated at some positive integers x          
equivalent conjecture:
*   no zero occurs except the two marked ones        
K=   x=   1 2 3 4 5 6 7
0 f0(x)= -1 -1 -1 -1 -1 -1 -1
1 f1(x)= 0 1 2 3 4 5 6
2 f2(x)= -1 0 2 5 9 14 20
3 f3(x)= -3 -4 -2 5 19 42 76
4 f4(x)= -7 -18 -28 -25 9 98 272
5 f5(x)= -15 -64 -158 -271 -317 -126 580
6 f6(x)= -31 -210 -748 -1825 -3351 -4606 -3760
7 f7(x)= -63 -664 -3302 -10735 -26141 -50478 -77324
8 f8(x)= -127 -2058 -14068 -59425 -183111 -446782 -896848
... ... ...
                       
  first approach:                
can the polynomials f3(x)...foo(x) (or F3(x)..Foo(x) )be proven to be irreducible over Z+?
                       
  second approach:                
derive a proof, that all real roots for f3(x)..foo(x) are irrational, exploiting the heuristic evidence
    for evidence: see page "real roots..."