Wolfram Cloud Document

Primality of

∑

k=1

1+k

(-1)

by Vitaliy Kaurov

In a thought-provoking LinkedIn group called Fermat’s Library I saw a post about primality of the following expression

In[]:=

Inactivate[Sum[

1+k

(-1)

k!,{k,1,n}],Sum]%//TeXForm

Out[]=

∑

k=1

1+k

(-1)

Out[]//TeXForm=

\underset{k=1}{\overset{n}{\sum }}(-1)^{k+1} k!

But surprisingly they counted only up to 15. So lets generalize. Define a function

In[]:=

f[n_]:=Sum[

1+k

(-1)

k!,{k,1,n}];

After automatic symbolic simplification the closed form is expressed in special function and no explicit summation is required

In[]:=

f[n]//TraditionalForm

Out[]//TraditionalForm=

(-

(-1)

)

n+2

(1)Γ(n+2)+Ei(-1)+1

◼

n+2

(1)

- exponential integral function

◼

Ei(-1) - exponential integral function

◼

Γ(n+2) - Euler gamma function

Out of first 1000 n

In[]:=

data=ParallelTable[{n,PrimeQ[#],#}&@f[n],{n,1,1000}];

the following result into prime f[n]

In[]:=

Select[data,#[[2]]==True&][[All,1]]

Out[]=

{3,4,5,6,7,8,10,15,19,41,59,61,105,160,661}

This is an OEIS number. Here is the table for all up to the n=41 (the two next ones after n=15):

In[]:=

TableForm[data[[;;41]],TableHeadings{None,{"n","Prime Test","f[n]"}}]

Out[]//TableForm=

n	Prime Test	f[n]
1	False	1
2	False	-1
3	True	5
4	True	-19
5	True	101
6	True	-619
7	True	4421
8	True	-35899
9	False	326981
10	True	-3301819
11	False	36614981
12	False	-442386619
13	False	5784634181
14	False	-81393657019
15	True	1226280710981
16	False	-19696509177019
17	False	335990918918981
18	False	-6066382786809019
19	True	115578717622022981
20	False	-2317323290554617019
21	False	48773618881154822981
22	False	-1075227108896452857019
23	False	24776789629988523782981
24	False	-595671612103250915577019
25	False	14915538431227735068422981
26	False	-388375922695377900515577019
27	False	10500493527722974260252422981
28	False	-294387851083990886241251577019
29	False	8547374142655711068302364422981
30	False	-256705485669535347568006115577019
31	False	7966133168508387470157556764422981
32	False	-255164703765185142697060455395577019
33	False	8428152915046701352821133945884422981
34	False	-286804646124557439494797475697635577019
35	False	10046343320261587490171853861825564422981
36	False	-361946983469639629977827594289009635577019
37	False	13401806107756705416338151987291892764422981
38	False	-509620811358844406343669072112782398435577019
39	False	19888261269838598952296612667790114958364422981
40	False	-796027021978059135393314656928325779313635577019
41	True	32656499591185747972776747396512425885838364422981