The Last Fermat theorem

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F-sums in sequences

Now take a more general problem. Let us search all (arithmetic) sequences f= {f(i)} that have such members f(r),f(b),f(c), which satisfy so called F-sum: f(r)+f(b)=f(c).

Fermat’s theorem reduces field of sequences to f(n) = nk (k is constant), which means that F-sums ak +bk =ck are observed.

Basic terms

Arithmetical sequences

Sequence {f(t)} is called arithmetical sequence of k-th order, if its k-th differential sequence is constant.

Arithmetical sequences are results of algebraic functions with constant exponents (linear and power functions,...).
If {d(t)} is differential sequence of sequence {f(t)}, then {f(t)} is sum sequence of sequence {d(t)}.

Characteristics

Let us call the set of first numbers of particular differential sequences (starting with constant sequences)

characteristic of sequences.

Any sequence with given leadingelement has just one characteristic.

E.g. sequence or third powers:

0, 1, 8,  27, 64, 125, 216, 343, 512, 729, ...
  1,  7, 19, 37, 61, 91, 127, 169, 217, ...
    6,  12, 18, 24, 30, 36, 42, 48, ... 
       6,  6, 6, 6, 6,  6,  6,  6, ...

is of third order with characteristic [6,6,1,0].

We write characteristic from below, i.e. beginning with constant sequences. But, to simplify some next formulas, let us index it from above. Elements of characteristics {c(h)} are then... c2,c1,c0; in our example [6,6,1,0] is c0=0,c1=1,c2=6,c3=6.

Sequence D, that results from sum (differences) of characteristic KD= KA + KB of sequences A and B, is equal to sum of these sequences,

i.e. D=A+B.

Sum of characteristics of sequences of different orders must be done from the right with missing left position completed by zeros.
E.g. the sum of characteristic [6,6,1,0] of sequence A=f(t) = t³ and [2,1,0] of sequence B=b(t) = t² is characteristic KD= [6,8,2,0] of sequence D=f(t)+g(t)=t³+t²:

    Charakter.    Sequence  
    ─────────────────────────────────────────────
CA= [6,6,1,0]    A=  {0,1, 8,27,64,125,216,... }
CB= [0,2,1,0]    B=  {0,1, 4, 9,16, 25, 36,... }
    ─────────────────────────────────────────────
CD= [6,8,2,0]    D=  {0,2,12,36,80,150,252,... }

Elementary sequences

Characteristics of basic sequences 
for k=0..5: 
     • k=0: [1] 
     • k=1: [1,c0] 
     • k=2: [1,0,c0] 
     • k=3: [1,0,0,c0] 
     • k=4: [1,0,0,0,c0] 
     • k=5: [1,0,0,0,0,c0] 

Zero sequence , i.e. sequence {0,0,0,0,0,...} has all differential sequences filled by zero. Therefore it can represent all sequences of any order, i.e. sequences with characteristics [0],[0,0],[0,0,0], and so on.

We call basic sequence of order k the sequence s with characteristic of the form:

[1,0,0...,0,c(0)], i.e. with constant 1 of the last difference sequence (i.e. c(k)=1) and with zero shift of the next differential sequences (i.e. for h ε <1,k−1> is c(h)=0). Individual basic sequences are distinguished by the number c(0). The basic sequences are easy to read and assemble.

For each F-sum (a, b, c) can be of zero sequence created a basic sequence [1,0, ... 0, ck]. The zero sequence reflects an infinite number of elementary sequences.

Characteristics of power sequences 
for k=0..5: 
    • k=0: [1] 
    • k=1: [1,0] 
    • k=2: [2,1,0] 
    • k=3: [6,6,1,0] 
    • k=4: [24,36,14,1,0] 
    • k=5: [120,240,150,30,1,0]           

In power sequences {f(t)} =tk is c(k)=k! and for k>0 is c(1)=1 and c(0)=0. For c(2) is (0+1)∙2=2, (2+1)∙2=6, (6+1)∙2=14, (14+1)∙2=30, i.e. coefficients c(2) make sequence defined by recurrent rule f(k+1)=(f(k)+1)∙2.

Generally it holds:

c(m) = f_sum_t_0-m f_binom04 ∙(t−m)k = f_sum_t_0-m (−1)k f_binom04 ∙(m−t)k

When m>k is c(m)=0. f_binom05
E.g. for k=4, m=0..4:
c(0)= f_binom_0_0 ∙04=0

c(1)= f_binom_1_0 ∙14 f_binom_1_1 ∙04=1−0=1

c(2)= f_binom_2_0 ∙24 f_binom_2_1 ∙14+ f_binom_2_2 ∙04=16−2+0=14

c(3)= f_binom_3_0 ∙34 f_binom_3_1 ∙24+ f_binom_3_2 ∙14 f_binom_3_3 ∙04=81−48+3−0=36

c(4)= f_binom_4_0 ∙44 f_binom_4_1 ∙34+ f_binom_4_2 ∙24 f_binom_4_3 ∙14+ f_binom_4_4 ∙04=256−324+96−4+0=24
─────────────────────────────────────────────
For m=5 is:
c(5)= f_binom_5_0 ∙54 f_binom_5_1 ∙44+ f_binom_5_2 ∙34 f_binom_5_3 ∙24+ f_binom_5_4 ∙14 f_binom_5_5 ∙04
=625−1280+810−160+5−0= 0

Because if m=k is c(k)=k!, then c(k)=∑ f_binom27 ∙(t−k)k = k!

Equivalent sequences

Sequences are called equivalent, if they are identical, except indexes of their members. E.g. sequences with characteristics [6,6,1,0] and [6,0,1,−1] are equivalent.

Charakter.      Sequence  
─────────────────────────────────────────────
[6,6,1,0]       { 0,1,8,27,64,125,216,...  }
[6,0,1,−1]      {−1,0,1, 8,27, 64,125,216,.}

More generally – (in this example for m=0, resp.−1) all sequences [6,6(m+1),3m²+3m+1, m³] are equivalent.

F-sequences

If there exists numbers f(a),f(b),f(c) in sequence {f(n)} = f(0),f(1),f(2),... such, that it holds f(a)+f(b)=f(c), we speak of F-sum.

F-sum is identified by F-indexes (a,b,c), 1≤ a < b < c.

In the following table sequences with characteristics [6,6,1,0..8] are written,

together with their first existing F-sum (in first 100 members of sequences) and its F-indexes.

Charakter.  Sequence           F-indexes    F-sum
──────────────────────────────────────────────────────────
[6,6,1,0] {0,1,8,27,64,125,… }
[6,6,1,1] {1,2,9,28,65,126,… } ( 6, 8, 9)   217+513=730
[6,6,1,2] {2,3,10,29,66,127,…} ( 5, 6, 7)   127+218=345
[6,6,1,3] {3,4,11,30,67,128,…}
[6,6,1,4] {4,5,12,31,68,129,…}
[6,6,1,5] {5,6,13,32,69,130,…}
[6,6,1,6] {6,7,14,33,70,131,…} (43,58,65)   79513+…=274631
[6,6,1,7] {7,8,15,34,71,132,…} ( 0, 1, 2)   7+8=15
[6,6,1,8] {8,9,16,35,72,133,…} (12,16,18)   1736+4104=5840

E.g. in sequence {7,8,15,34,71,132,223,...} is third member (15) sum of previous two members (7+8).

We call F-sequences the sequences that have F-sum.

Multiples of sequences

Let sequence f'(n) is q multiple of sequence f(n), i.e.f'(n)=q∙f(n).

Characteristic c' of sequence f'(n) is q multiple of characteristics c of sequence f(n),

i.e. c'(m)=q∙c(m), mεN,qεN.

If there exists F-sum with indexes (a,b,c) in f(n), then there exists analogous F-sum (with the same indexes) in sequences f'(n).

E.g. F-sum with indexes (0,2,5) exists in all sequences with characteristics [1,0,9], [2,0,18], [3,0,27], [4,0,36],...

Charakter.  Sequence                  F-indexes  F-sum
───────────────────────────────────────────────────────────
[1,0,9]    {9,9,10,12,15,19,24,...}  ( 0, 2, 5)  9+10=19
[2,0,18]   {18,18,20,24,30,38,48,.}  ( 0, 2, 5) 18+20=38
[3,0,27]   {27,27,30,36,45,57,72,.}  ( 0, 2, 5) 27+30=57
[4,0,36]   {36,36,40,48,60,76,96,.}  ( 0, 2, 5) 36+40=76

If sequence f(n) is F-sequence, then all sequences a'(n) are also F-sequences, for qεN.

Sequences having equal F-indexes

To given F-indexes (a,b,c) entire class of sequences (subset of all sequences) exists with necessary F-sums.

In other words: for each F-sum of a given sequence there exists always an analogous F-sum also in others sequences.

The following assertions are obvious:

Zero sequence contains all possible F-sums. There exists sequences, that contains infinite number of F-sums (e.g. sequence f(n) = n²). There exists sequences, that have no F-sum (e.g. sequence f(n) = n³).

Further:

Two given F-sums can exist also in more sequences. Thus we may not always get two different sequences from the transformation of two different F-sums from one sequence. E.g. F-sums with indexes (6,8,10) and (5,12,13) exist in both the sequences: with characteristics [2,1,0] and [1,0,2].
Charakter. Sequence    
─────────────────────────────────────────────────────────
[2,1,0]    {0,1,4,9,16,25,36,49,64,81,100,121,144,169..}
[1,0,2]    {2,2,3,5, 8,12,17,23,30,38, 47, 57, 68, 80..}

In [2,1,0] is 36+64 = 100 and in [1,0,2] is for the same indexes 17+30=47. Similarly is 25+144 = 169 and 12+68=80.

To given F-sum there exists one equivalent in zero sequences and one equivalent in certain basic sequence.

Two sequences have F-sums with the same indexes (a,b,c), if it holds for characteristics c(m) and c'(m):

∑Δc(m)∙( f_binom_cbam) = 0; (m=0..k)

where Δc(m) = c'(m)−c(m) and f_binom28 , f_binom29 , f_binom30 are binomial coefficients.

Because(f_binom_cba0 ) = 1−1−1 = −1,we can evaluate c(0) from the previous formula:

∑Δc(0)= ∑Δc(m)∙( f_binom_cbam); (m=1..k)

For m=1 it holds Δc(0)=Δc(1)∙(c−b−a). For (0,2,5) is c−b−a= 5−2−0=3.

So F-sum with indexes (0,2,5) exists also in sequences [1,1,12],[1,2,15],..., [2,1,21],[2,1,24], and so on.


Charakter. Sequence                  F-indexes    F-sum
──────────────────────────────────────────────────────────
[1,1,12]  {12,13,15,18,22,27,33,..}  ( 0, 2, 5)   12+15=27
[1,2,15]  {15,17,20,24,29,35,42,..}  ( 0, 2, 5)   15+20=35
[1,3,18]  {18,21,25,30,36,43,51,..}  ( 0, 2, 5)   18+25=43
[1,4,21]  {21,25,30,36,43,51,60,..}  ( 0, 2, 5)   21+30=51
────
[2,1,21]  {21,22,25,30,37,46,57,..}  ( 0, 2, 5)   21+25=46
[2,2,24]  {24,26,30,36,44,54,66,..}  ( 0, 2, 5)   24+30=54
[2,3,27]  {27,30,35,42,51,62,75,..}  ( 0, 2, 5)   27+35=62
─────
[3,1,30]  {30,31,35,42,52,65,81,..}  ( 0, 2, 5)   30+35=65
[3,2,33]  {33,35,40,48,59,73,90,..}  ( 0, 2, 5)   33+40=73

Let us try to deduce (according to previous rules) basic sequences, that have F-sums analogous to selected Pythagorean F-sums, i.e. F-sums (3,4,5), (6,8,10) and (5,12,13), from power sequences of 2.order.

Let us look for such c0, that respect relation of transformation from sequence with characteristic Cm=[2,1,0] to sequences with characteristic Cz=[1,0,c0].

Difference of these two characteristics is ΔC=[1,0,c0]−[2,1,0]=[−1,−1,c0]
It holds:
Δc0=Δc2∙( f_binom_cba2 )+Δc1∙ ( f_binom_cba1 ).

After substitution we get for a=3,b=4,c=5: Δc0=+1, for a=6,b=8,c=10:Δc0=+2 and for a=5,b=12,c=13 also: Δc0=+2.
Therefore basic sequences for F-sum with indexes (3,4,5) is: [1,0,1] and for F-sums with indexes (6,8,10) and (5,12,13): [1,0,2].

Charakter.  Sequence                               F-sum
──────────────────────────────────────────────────────────
[2,1,0]   {0,1,4,9,16,25,36,49,64,81,100,121,…}    9+16= 25
[1,0,1]   {1,1,2,4,7,11,16,22,29,37,46,56,67,…}    4+7= 11
───────────────────────────────────────────────────────────
[2,1,0]   {0,1,4,9,16,25,36,49,64,81,100,121,…}    36+64=100
[1,0,2]   {2,2,3,5,8,12,17,23,30,38, 47, 57,… }    17+30= 47
───────────────────────────────────────────────────────────
[2,1,0]   {0,1,4,9,16,25,36,…,81,100,121,144,…}    25+144=169
[1,0,2]   {2,2,3,5,8,12,17,23,30,38,47,57,68,…}    12+ 68=80
───────────────────────────────────────────────────────────

Basic sequences

Basic sequence order k has F-sum with indexes (a,b,c) if c0= f_binom_c_k f_binom_b_k f_binom_a_k .
It follows directly from transformation of sequences [0,0,0,...,0] to [1,0,0,...,c0]

E.g. F-sum with indexes (0,2,5) exists in sequences with characteristic [1,0,9], because c0= f_binom_5_2 f_binom_2_2 f_binom_0_2 = 10−1−0 = 9.

Charakter. Sequence                   F-indexes    F-sum
──────────────────────────────────────────────────────────
[1,0,9]    {9,9,10,12,15,19,24,...}  ( 0, 2, 5)    9+10=19

Binomial sequences

Sequence with characteristic [1,0,0] has members f(a)= f_binom36 .

Charakter. Posloupnost               F-indexy      F-součet
───────────────────────────────────────────────────────────
[1,0,0]    {0,0,1,3,6,10,15,21,...}  ( 4, 6, 7)    6+15=21

There is a Fermat-like question, for which (a,b,c) in sequences of order k exist F-sums f_binom_a_k + f_binom_b_k = f_binom_c_k ?

For k=2 to 5 we get experimentally these indexes:

k=2: (4,6,7),(5,10,11),(6,15,16),..(6,7,9),(7,10,12),(10,22,24),(11,27,29),... (9,11,14),(10,14,17),(12,21,24),...(12,15,19), ... k=3: (10,16,17),(22,56,57),(32,57,60),(36,120,121), ... k=4: (132,190,200),..? k=5: ?

E.g. f_binom_5_2 + f_binom44 = 10 + 45 = 55 = f_binom45 .

Indexes in the first row are of the form (n, f_binom46 , f_binom46 +1).

Power sequences

Let us check on simple example of sequences of 2.order, that in power sequences [2,1,0] corresponds indexes (a,b,c) to F-sum a²+b²=c².
By transformation from [0,0,0] to [2,1,c0] we get:
c0 = 2∙(
f_binom_cba2) + 1∙( f_binom_cba1 )

= 2∙t(c−1)/2−2∙s(b−1)/2−2∙r(a−1)/2 + 1∙(c−b−a) = c²−b²−a²
In case c0=0 is therefore a²+b²=c².

Similarly in sequences of 3.order transformation from [0,0,0,0] to [6,6,1,c0] provides:
c0 =6∙(
f_binom_cba3 ) + 6∙( f_binom_cba2) + 1∙( f_binom_cba1 ).
For c is 6∙t(c−1)(c−2)/6 + 6∙t(c−1)/2 + 1∙(c) = c³−3l²+2l + 3l²−3l+c = c³.
Proto when c0=0, is a³+b³=c³.

Shift of sequence

Change of characteristics c(m) by value v make shift of each member of the sequence f(n) by value f_binom50 ∙v.

We call the value m order of the shift. E.g. for k=3.

Members f(n) sequences [6,6,1,v] are with regard to k sequences [6,6,1,0] shifted o value v∙ f_binom51 = v:

Charakter. Sequence                  F-indexes  F-sum
─────────────────────────────────────────────────────────
[6,6,1,0]  {0,1, 8,27,64,125,216,..}
[6,6,1,1]  {1,2, 9,28,65,126,217,..} ( 6,8,9)   217+513=730
[6,6,1,2]  {2,3,10,29,66,127,218,..} ( 5,6,7)   127+218=345

Similarly are also members f(n) sequences [6,6,v,0] with regard to k sequences [6,6,0,0] shifted by value v∙ f_binom52 = v∙n:

Charakter. Sequence             F-indexes   F-sum
──────────────────────────────────────────────────────────
[6,6,0,0] {0,0, 6,24,60,120,…} ( 9,15,16)   720+3360=4080
[6,6,1,0] {0,1, 8,27,64,125,…}
[6,6,2,0] {0,2,10,30,68,130,…} (36,37,46)   46692+50690=97382

And members f(n) of sequences [6,v,0,0] with regard to k sequences [6,0,0,0] shifted by value v∙ f_binom46 :

Charakter. Sequence          F-indexes    F-sum
──────────────────────────────────────────────────────────
[6,0,0,0] {0,0,0,6,24,60,… } ( 0, 1, 2)   0+0=0
[6,1,0,0] {0,0,1,9,30,70,… } (11,19,20)   1045+5985=7030
[6,2,0,0] {0,0,2,12,36,80,…} (31,37,43)   27900+47952=75852

Shifts of power sequences

Shift of order 0: F-sum f(a)+f(b)=f(c) with F-indexes (a,b,c) exists in power sequences shifted by v,

i.e. {f(n)} = {nk+v}, if v = ck−bk−ak.
E.g. for k=2 and indexes (1,2,3) we get v=3²−2²−1² = 9−4−1=4.

Therefore F-sum with indexes (1,2,3) appears in sequences with characteristic [2,1,4]:

Charakter. Sequence                    F-indexes  F-sum
──────────────────────────────────────────────────────────
[2,1,0]    {0,1,4,9,16,25,36,49,... }  (3,4,5)    9+16=25
[2,1,1]    {1,2,5,10,17,26,37,50,...}  (4,8,9)    17+65=82
[2,1,2]    {2,3,6,11,18,27,38,51,...}  (3,5,6)    11+27=38
[2,1,3]    {3,4,7,12,19,28,39,52,...}  (0,1,2)    3+4=7
[2,1,4]    {4,5,8,13,20,29,40,53,...}  (1,2,3)    5+8=13
[2,1,5]    {5,6,9,14,21,30,41,54,...}  (0,2,3)    5+9=14

Shift of order 1: F-sum f(a)+f(b)=f(c) with F-indexes (a,b,c) exists in sequences shifted by v∙n, i.e. {f(n)} = {nk+v∙n}, if:

v = (ck−bk−ak)/(c−b−a)

For k=3 and indexes (2,3,4) we get v= 4³−3³−2³ = 64−27−8 = 29.

Therefore F-sum with indexes (2,3,4) appears in sequences with characteristic [6,6,30,0]:

n          0   1   2   3    4    5    6
────────────────────────────────────────
[6,6,1,0]  0   1   8  27   64  125  216 
29∙n       0  29  58  87  116  145  174 
────────────────────────────────────────
[6,6,30,0] 0  30  66 114 180   270  390    66+114 = 180

Generalization:
Let us consider change of sequence after shift of their m-th differential sequence by value v.

Then in the new sequence are all the original members shifted by v∙ f_binom50 , i.e. {f(n)}={nk+v∙ f_binom50 }.

In this new sequence there exists F-sum f(a)+f(b)=f(c) with indexes (a,b,c) if:

f_binom_expr_v

Qualification of characteristics of F-sums

We are interested, what properties must have sequences, in order to have any F-sum.

Some cases are trivial; e.g. sequences of 0.order never have F-sum, except the case c0=0.

I.e. in sequence 5,5,5,5,5,5,5.. can never appear an other number like 5+5 = 10, ....

In other cases we will observe influences of differential sequences existence of F-sums.

Sequences of 1.order

Any sequence of 1.order can be written in the form:
f(n) = c1∙n+ c0; where c0 = c1(c−b−a). So F-sum exists, if number c0 is divisible by number c1 (with no remainder):

Charakter. Sequence                  F-indexes    F-sum
───────────────────────────────────────────────────────────
[1,0]   {0,1,2,3,4,5,6,7,8,9,... }  ( 1, 2, 3)    1+2=3
[1,1]   {1,2,3,4,5,6,7,8,9,10,...}  ( 0, 1, 2)    1+2=3
[1,2]   {2,3,4,5,6,7,8,9,10,...  }  ( 0, 1, 3)    2+3=5
[1,3]   {3,4,5,6,7,8,9,10,11,... }  ( 0, 1, 4)    3+4=7
───────────────────────────────────────────────────────────
[2,0]   {0,2,4,6,8,10,12,14,...  }  ( 1, 2, 3)    2+4=6
[2,1]   {1,3,5,7,9,11,13,15,...  }
[2,2]   {2,4,6,8,10,12,14,16,... }  ( 0, 1, 2)    2+4=6
[2,3]   {3,5,7,9,11,13,15,17,... }
───────────────────────────────────────────────────────────
[3,0]   {0,3,6,9,12,15,18,21,... }  ( 1, 2, 3)    3+6=9
[3,1]   {1,4,7,10,13,16,19,22,...}
[3,2]   {2,5,8,11,14,17,20,23,...}
[3,3]   {3,6,9,12,15,18,21,24,...}  ( 0, 1, 2)    3+6=9

In sequences of 1.order can (potentially) exist F-sums only if:

c0 ≡ 0 (mod c1)

Sequences of 2.order

In sequences of 2.order we find similar formula:

c0≡0 mod (c1,c2)

where (c1,c2) is greatest common divisor of numbers c1 and c2.

E.g. for c1 = 6 we can distinguish following cases:

(c1,c2) F-sums exists for
────────────────────────────────────
(6,1)=1  all sequences
(6,2)=2  c0≡0 mod 2
(6,3)=3  c0≡0 mod 3
(6,4)=2  c0≡0 mod 2
(6,5)=1  all sequences
(6,6)=6  c0≡0 mod 6

Therefore:

In sequences [6,8,c0] F-sum exists if c0≡0 mod (6,8), i.e. when c0 is even:
Charakter. Sequence                 F-indexes   F-sum
──────────────────────────────────────────────────────────
[6,8,0]  {0,8,22,42,68,100,138,...} ( 4,10,11)  68+350=418
[6,8,1]  {1,9,23,43,69,101,139,...}
[6,8,2]  {2,10,24,44,70,102,140,..} ( 3, 6, 7)  44+140=184
[6,8,3]  {3,11,25,45,71,103,141,..}
[6,8,4]  {4,12,26,46,72,104,142,..} ( 2, 3, 4)  26+46= 72
[6,8,5]  {5,13,27,47,73,105,143,..}
[6,8,6]  {6,14,28,48,74,106,144,..} ( 4,11,12)  74+424=498
In sequences [6,6,c0] F-sum exists only for c0≡0 mod 6:
Charakter. Sequence                 F-indexes    F-sum
──────────────────────────────────────────────────────────
[6,6,0]  {0,6,18,36,60,90,126,... } ( 3, 5, 6)   36+90=126
[6,6,1]  {1,7,19,37,61,91,127,... }
[6,6,2]  {2,8,20,38,62,92,128,... }
[6,6,3]  {3,9,21,39,63,93,129,... }
[6,6,4]  {4,10,22,40,64,94,130,...}
[6,6,5]  {5,11,23,41,65,95,131,...}
[6,6,6]  {6,12,24,42,66,96,132,...} ( 2, 3, 4)   24+42=66
In sequences [6,7,c0] F-sum exists for all c0:
Charakter.  Sequence               F-indexes   F-sum
──────────────────────────────────────────────────────────
[6,7,0]  {0,7,20,39,64,95,132,..}  ( 7,28,29)  175+2464=2639
[6,7,1]  {1,8,21,40,65,96,133,..}  ( 6,21,22)  33+1408=1541
[6,7,2]  {2,9,22,41,66,97,134,..}  ( 5,15,16)  97+737= 834
[6,7,3]  {3,10,23,42,67,98,135,.}  ( 4,10,11)  67+343= 410

More general criterion

Let us write more general relation for F-sum f(a)+f(b)=f(c) with help of characteristic of sequences.

E.g.
f(a)=c0 f_binom33 + c1 f_binom39 + c2 f_binom36 + … = c0+ c1∙r+ c2∙r(a−1)/2+…

After an arrangement, an expressions dependent on (c−b−a),(c²−b²−a²) ... appear

To simplify these expressions, let use define function d(m):

d(m) = cm−bm−am

E.g. for sequence of the 3. order we get relation:
c0 + c1∙d1 + c2(d2−d1)/2 + c3(d3−3d2−2d1)/6 = 0

If some number divide numbers c1,c2,c3,... it must divide also the number c0, otherwise the previous relation is impossible.

F-sums can exist only in sequences if it holds:

c0≡ 0 mod (c1,c2,c3,...)

Zero members c1,c2,c3,... have to be skipped in calculation of greatest common divisor (c1,c2,c3,...).

In sequences with characteristic [2,0,1] can not exist F-sum, because it does not hold 1 ≡ 0 mod 2:
Charakter.   Sequence               Note 
───────────────────────────────────────────────────────
[2,0,1] {1,1,3,7,13,21,31,43,...}   f(n) = n²−n+1

But to validity of the Fermat’s theorem this criterion brings nothing new. For sequences f(k) = ak is always c0 = 0.

Qualification of indexes of F-sums

Transformation to zero sequences

Zero sequence contains F-sums with any indexes. Therefore, if F-sums exist in some given sequence, there exist F-sums also in the corresponding zero sequence.

And (vice versa) in sequences can exist F-sum with indexes (a,b,c) only if it can be derived from zero sequences by transformation for given (a,b,c).

E.g. in sequences [2,0,1] no F-sum can exists, because there does not exists any (a,b,c) for transformation from [0,0,0] to [2,0,1].

For existence of such transformation is needed:

2∙( f_binom34 f_binom35 f_binom36 )=1, i.e. (c²−c) −(b²−b) −(a²−a)=1. But number m²−m is even for all mεZ.

But these relations are not distinct from those, that was deduced with help of breakdown of f(a)+f(b)=f(c) in previous paragraphs,

so they do not provide nothing new about power sequences.
E.g. transformation [0,0,0] to [2,1,0] for (a,b,c) leads to c²−a²−b²=0, which is evident.

Transformation to basic sequences

Let us try to find basic sequences, that have F-sums with indexes equal to indexes in given power sequences.

Let us assume, that if basic sequence to given sequences having F-sum with indexes (a,b,c) does not exists,

then in given sequence the F-sum (a,b,c) is not possible.

In sequences of the 2.order it follows from transformation of [2,1,0] to [1,0,c0]:
2∙c0 = (c²−b²−a²) − (c−b−a) = d2−d1

Because simultaneously d2=0 (i.e. a²+b²=c²) then:
2∙c0 = −d(1) = −d(1) = −(c−b−a) = a+b−c

Therefore it must hold for indexes F-sums of power sequences of 2.order:
−d1 ≡0 mod 2 i.e. j+i−c≡0 mod 2

This formula holds in Pythagorean triangles, e.g. 3+4−5≡0 mod 2, 5+12−13≡0 mod 2, ...

We will find similar formulas for higher orders:

Order  Formula
───────────────────────────────────────────
2    −d1  ≡0 mod 2  
3    −3∙d2+2∙d1 ≡ 0 mod 6
4    −6∙d3+11∙d2−6∙d1 ≡ 0 mod 24
5    −10∙d4+35∙d3−50∙d2+24∙d1 ≡ 0 mod 120

Numbers u of particular d(m) are Stirling’s numbers of the first case s(k,m), see Recurrent sequences.

More general criterion

To any power sequence of k-th order with indexes of F-sum (a,b,c)

there always exists one basic sequence with characteristic [1,0,...,0,c0], if it holds k!∙c0 = ∑ s(k,m)∙d(m), where d(m)=cm−bm−am.

Therefore it must hold for indexes of F-sum (a,b,c) in power sequences k-th order:

∑ s(k,m)∙d(m) ≡ 0 (mod k!)

In opposite case there corresponding basic sequence does not exists.

E.g. indexes (1,2,4) can not be solution of a³+b³=c³ because:
d1= c−b−a= 4−2−1 = 1, d2= c²−b²−a²= 16−4−1= 11, −3∙d2+2∙d1 = −33+2 = −31

and −31 is not divisible by 6 (i.e. not congruent to 0 according to module k!=3!=6).

We can rewrite the said relations also to the form:
δc− δb− δa ≡0 mod k! where δa = a(a−1)(a−2)..(a−k+1).

F-sums - special cases

We will follow some special cases of relations described above in the part F-sums in seuences in the following paragraphs.

We are looking for all sequences f= {f(i)} having such members f(a),f(b),f(c),

that so called F-sum f(a)+f(b)=f(c) holds.

We are interested in how these relations ( in case of sequence f(n) = nk ) influence on the posibility of solution ak +bk =ck (Last Fermat theorem says, that such solution do not exists).

Let us focus on the case k=3 first. We observe arithmetic sequences of the 3-th order ( 3-th differential sequence is constant ).

Such sequence is e.g.

{0,18,42,78,132,210,... }

 0,  18,   42,   78,   132,  210, ...   given sequence {f(t)}
    18,  24,   36,    54,  78, ...      1. differential {d1(t)}
        6,    12,   18,   24,   ...     2. differential {d2(t)}
           6,    6,    6,  ...          3. differential {d3(t)} 

This sequence has characteristic

[6,6,18,0] (number on the left) - c0=0,c1=18,c2=6,c3=6.

In this sequence it holds

78+132=210, so, we say, there exists F-sum

(3,4,5) ~ f(3) +f(4) = f(5) .

On the contrary we know that in sequence {0,1, 8,27,64,125,216,... } with characteristic [6,6,1,0] no such F-sum exists.

The prove (for k=3) was already done - firstly by L. Euler (1707-1783) and J.L. Lagrange (1736-1813).

What is the distinction of the two sequences ( [6,6,18,0] and [6,6,1,0])?

When there exists some F-sum and when not?

Sequences [6,6,C,0]

In the following outline there are sequences with characteristics [6,6,C,0], where C=1,2,..30, their firs existing F-sum (in first 100 members of each sequence) and coresponding F-indexes.

Charakter.   Sequence                      F-indexes    F-sum
──────────────────────────────────────────────────────────────────────────
[6,6,1,0]     {0,1,8,27,64,125,216,... }
[6,6,2,0]     {0,2,10,30,68,130,222,...}  ( 36, 37, 46) 46692+50690=97382
[6,6,3,0]     {0,3,12,33,72,135,228,...}  ( 10, 12, 14) 1020+1752=2772
[6,6,4,0]     {0,4,14,36,76,140,234,...}  ( 10, 18, 19) 1030+5886=6916
[6,6,5,0]     {0,5,16,39,80,145,240,...}  ( 72, 74, 92) 373536+405520=779056
[6,6,6,0]     {0,6,18,42,84,150,246,...}  (  8, 13, 14) 552+2262=2814
[6,6,7,0]     {0,7,20,45,88,155,252,...}  ( 13, 27, 28) 2275+19845=22120
[6,6,8,0]     {0,8,22,48,92,160,258,...}  ( 29, 63, 65) 24592+250488=275080
[6,6,9,0]     {0,9,24,51,96,165,264,...}  ( 20, 24, 28) 8160+14016=22176
[6,6,10,0]    {0,10,26,54,100,170,...  }  (  4,  5,  6) 100+170=270
[6,6,11,0]    {0,11,28,57,104,175,...  }
[6,6,12,0]    {0,12,30,60,108,180,...  }  (  5,  7,  8) 180+420=600
[6,6,13,0]    {0,13,32,63,112,185,...  }  ( 20, 36, 38)  8240+47088=55328
[6,6,14,0]    {0,14,34,66,116,190,...  }  ( 39, 80, 83) 59826+513040=572866
[6,6,15,0]    {0,15,36,69,120,195,...  }  ( 15, 34, 35) 3585+39780=43365
[6,6,16,0]    {0,16,38,72,124,200,...  }  (  8,  9, 11) 632+864=1496
[6,6,17,0]    {0,17,40,75,128,205,...  }
[6,6,18,0]    {0,18,42,78,132,210,...  }  (  3,  4,  5) 78+132=210
[6,6,19,0]    {0,19,44,81,136,215,...  }  ( 30, 36, 42) 27540+47304=74844
[6,6,20,0]    {0,20,46,84,140,220,...  }  ( 12, 13, 16) 1956+2444=4400
[6,6,21,0]    {0,21,48,87,144,225,...  }  ( 13, 28, 29) 2457+22512=24969
[6,6,22,0]    {0,22,50,90,148,230,...  }  (  4,  6,  7) 148+342=490
[6,6,23,0]    {0,23,52,93,152,235,...  }
[6,6,24,0]    {0,24,54,96,156,240,...  }  (  5,  8,  9) 240+696=936
[6,6,25,0]    {0,25,56,99,160,245,...  }  ( 26, 54, 56) 18200+158760=176960
[6,6,26,0]    {0,26,58,102,164,250,... }
[6,6,27,0]    {0,27,60,105,168,255,... }  ( 13, 15, 18) 2535+3765=6300
[6,6,28,0]    {0,28,62,108,172,260,... }  ( 30, 54, 57) 27810+158922=186732
[6,6,29,0]    {0,29,64,111,176,265,... }
[6,6,30,0]    {0,30,66,114,180,270,... }  (  2,  3,  4) 66+114=180

We will derive relation for C as function a,b,d: C=F(a,b,d) from observation F-sums (a, b, b+d) for d=1,2,... :

For d=1:

    Charakter.     Sequence         F-indexes     F-sum
────────────────────────────────────────────────────-─────────────────────
[6,6,30,0] {0,30,66,114,180,270,... } ( 2, 3, 4) 66+114=180 [6,6,54,0] {0,54,114,186,276,390,...} ( 2, 4, 5) 114+276=390 [6,6,84,0] {0,84,174,276,396,540,...} ( 2, 5, 6) 174+540=714 ... [6,6,18,0] {0,18,42,78,132,210,... } ( 3, 4, 5) 78+132=210 [6,6,33,0] {0,33,72,123,192,285,... } ( 3, 5, 6) 123+285=408 ...

C = 3 * b(b+1)/(a-1) - a(a+1), for example when a=2, b=3 then C = 3*3*4/1 - 2*3 = 30, for a=3, b=5 is C = 3*5*6/2 - 3*4 = 33.

For d=2:
    Charakter.     Sequence      F-indexes     F-sum
    ────────────────────────────────────────────────────-─────────────────────
    [6,6,37,0]     {0,37,80,135,208,305,... } (  8, 10, 12) 800+1360=2160
    [6,6,60,0]     {0,60,126,204,300,420,...} (  8, 11, 13) 984+1980=2964
            ...

C = 6 * b(b+2)/(a-1) - a(a+2) - 3, for example when a=8, b=11 then C = 6*11*13/6 - 8*10 - 3 = 60


Let us mark c=b+d. In sequence with characteristic [6,6,C,0] there exists F-sum (a, b, c) = (a, b, b+d), when it holds:

C = 3*d*b*(b+d)/(a-d) - a(a+d) - d² + 1

Not existing F-sums

In sequences with characteristic [6,6,C,0] there exists F-sums (5,7,8), (5,8,9),(5,11,12), (5,12,13), it is (5, b, b+1) just for some b.

For example for b=9 such sums do not exists. It follows from the derived relation for a=5,b=9, d=1:

C = 3*d*b*(b+d)/(a-d) - a(a+d) - d² + 1 = 3*1*9*10/4 - 5*6 - 1 + 1 = 75/2 = 37.5

Not-existence of any F-sum is caused by fact, that expresion 3*d*b*(b+d) is not multiple of a(a+d),

so it do not hold: a(a+d) | 3*d*b*(b+d).


Power sequence

In sequence with characteristic [6,6,1,0] is C=1 . When it holds a³+b³=(a+d)³, then 1 = 3*d*b*(b+d)/(a-d) - a(a+d) - d² + 1, so

3*d*b*(b+d)/(a-d) - a(a+d) - d² = 0

By multiplication (a-d) we get:

3*d*b*(b+d) = a³- d³

3*d*b*c = a³- d³

From relation 3dbc=a³-d³ it follows, that d | a³ and therefore if the number a is prime (aεP), then there is d=1.

In general case ap +bp = cp (p εP) it holds: p < a < b < c < (a+b).

If b*c divide the expression a³-d³= (a-d) * (a²+ad+d2 ), then there is necessarily ( b*c, a-d) > 1 or b*c | a²+ad+d2 .

Catalan, Charles Eugene
Catalan, Charles Eugene [], 1814-1894, belgian matematician working in France and Belgium. In addition to number theory he was engaged also by descriptive geometry and combinatorics.

Catalan's formula for d=1

In case d=1 (a εN) i.e. 3b(b+1)= a³-1 have to be 3 | a³-1, what exclude 3 | a .

Because a³-1 = (a-1) (a²+a+1), there must be 3 | a-1 or 3 | a²+a+1; but the condition for the second is a=1 (mod 3),

so again 3 | a-1.

Charles E. Catalan has derived (y.1886) in general case ap +bp =(b+1)p (p εP), the following formula: