<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Prime Sums of Exactly Three Distinct Factorials</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">By</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Jonathan Vos Post</font></p>
<div class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">10-11 Aug 2006, revisions of 12-14 Aug 2006</font></div>
<div class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"></font> </div>
<div class="MsoNormal" style="MARGIN: 0in 0in 0pt">
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Acknowledgment:</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The author would like to acknowledge corrections and suggestions by Dr. George Hockney, Andrew Carmichael Post, Joshua Zucker, Gerald McGarvey, Franklin T. Adams-Watters,
<font color="#000000"><span class="sg">Hugo Pfoertner, Ralf Stefan, Stefan Steinerberger, </span>and Joseph Biberstine.</font></font></p></div>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">For the purpose of this note, we do NOT consider 0!=1 and 1!=1 to be distinct factorials.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The only prime factorial, of course, is 2!=2. The only way that the sum of two distinct factorials can be prime is if the sum is of the form n! + 1, the so-called "factorial primes"
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A002981 Numbers n such that n! + 1 is prime { 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, …}</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A088332 Primes of the form n!+1 {</font><tt><span style="FONT-SIZE: 10pt"> </span></tt><font face="Times New Roman">2, 3, 7, 39916801, 10888869450418352160768000001, …}. A! + B! cannot be prime for A>1, B>1, A=/=B because there would be at least one prime factor in common between A and B.
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">It is well-known that there can be prime sums of three or more factorials.<span style="mso-spacerun: yes"> </span>These are the prime subset of A059590 Sum of distinct factorials (0! and 1! not treated as distinct)
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">{</font><tt><span style="FONT-SIZE: 10pt"> </span></tt><font face="Times New Roman">0, 1, 2, 3, 6, 7, 8, 9, 24, 25, 26, 27, 30, 31, 32, 33, 120, …}, namely:
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A089359 Primes which can be partitioned into distinct factorials. 0! and 1! are not considered distinct {</font><tt><span style="FONT-SIZE: 10pt">
</span></tt><font face="Times New Roman">2, 3, 7, 31, 127, 151, 727, 751, 5167, …}. We are interested in a specific subset of that.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">We now tabularize and partially characterize Prime Sums of Exactly Three Distinct Factorials.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Lemma [trivial proof]:</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The prime sums of three distinct factorials must be of the form A! + B! + 1.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Lemma :</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">There are no solutions to "prime = 1! + 2! + n! for n>2."</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">[trivial proof]: For n>2, we have 3 | 1! + 2! + n! = 3 + n! and 1! + 2! + n! > 3.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The smallest solutions are:</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">4! + 3! + 1! = 31 is prime.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">5! + 3! + 1! = 127 is prime.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">6! + 3! + 1! = 727 is prime.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Lemma :</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">There are no solutions to "prime = 1! + 3! + n! for n>6."</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">[trivial proof]: For n>6, we have 7 | 1! + 3! + n! = 7 + n! and 7 + n! > 7.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">It is the case in general, for fixed A distinct from B, that A! + B! + 1! = prime has either no solutions, or several solutions with the upper bound for B being the least prime factor of
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A! + 1 = A051301(A).<span style="mso-spacerun: yes"> </span>[proof left to reader]. The number of such solutions for A = 1, 2, 3, 4, … is: 0, 0, 0, 3, 0, 0, 0, 9, 4, 3, 0, …
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Examples: </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">(a) There are no solutions to "prime = 1! + 4! + n! for n>4."</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The upper bound for B is LPF(1 + 4!) = least prime factor (25) = 5, so we need only check that 1! + 4! + 5! = 145 = 5 x 29 is divisible by 5, as is the case with all higher B.
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">(b) There are no solutions to "prime = 1! + 5! + n! for n>5."</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The upper bound for B is LPF(1 + 5!) = LPF(121 = 11^2) = 11.<span style="mso-spacerun: yes"> </span>We check for 5<B<11 and find that 1 + 5! + 6! = 841 = 29^2; that 1+ 5! + 7! = 5161 = 13 x 397; that 1+ 5! + 8! = 40441 = 37 x 1093; that 1+ 5! + 9! = 363001 = 17 x 131 x 163; that 1+ 5! + 10! = 3628921 = 67 x 54163; and we know that 1+ 5! + 11! = 39916921 = 11 x 3 628811 and all higher values of B give multiples of 11.
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">(c) There are no solutions to "prime = 1! + 6! + n! for n>6."</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The upper bound for B is LPF(1+ 6!) = LPF(721 = 7 x 103) = 7, so immediately we are preclused from solution as this and all higher values of B give a sum divisible by 7.
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The next smallest solutions to "prime = A! + B! + 1! for distinct A, B > 6" are:</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">8! + 7! + 1! = 45361 is prime.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">12! + 7! + 1! = 479006641 is prime.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Or, more compactly:</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">7! + n! + 1! is prime for n = {8, 12, 16, 23, 27, 33, 37, 42, 53}. We need check no further than 53<B<LPF(1 + 7! = 5041 =71^2) = 71, and we find no solutions in that range.
<span style="mso-spacerun: yes"> </span>Hence there are exactly those 9 solutions to B for "prime = 7! + B! + 1! for A<B."</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">I have used the "Alpertron" by Dario Alejandro Alpern to perform <span lang="EN" style="mso-ansi-language: EN">factorization using the Elliptic Curve Method for
</span>"prime = A! + B! + 1! for distinct A, B,<span style="mso-spacerun: yes"> </span>where 0 < A<= 50."</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">In that range, the hardest minimal solution is for A = 50, namely 50! + 111! + 1!, a prime of 181 digits. Coincidently, 181 is prime.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"></font> </p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">The following table of prime A! + B! + 1! Where, for convenience, we order A < B.<span style="mso-spacerun: yes"> </span>We also show, in the 2
<sup>nd</sup> column, the number of solutions if known, and in the 3<sup>rd</sup> column the least prime factor of A!+1 (i.e. the upper bound on B). For each nontrivial A, solutions have been searched for through B = 100</font>
</p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A! + B! + 1! is prime, A =/= B, A<B</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A <span style="mso-tab-count: 1"> </span>#solns<span style="mso-tab-count: 1"> </span>Bmax<span style="mso-tab-count: 1"> </span>B values in solutions
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">-------------------------------------------------------------------</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">1<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>2</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">2<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>3</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">3<span style="mso-tab-count: 1"> </span>3<span style="mso-tab-count: 1"> </span>7<span style="mso-tab-count: 1"> </span>
4, 5, 6</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">4<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>5</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">5<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>11</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">6<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>7</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">7<span style="mso-tab-count: 1"> </span>9<span style="mso-tab-count: 1"> </span>71<span style="mso-tab-count: 1"> </span>
8, 12, 16, 23, 27, 33, 37, 42, 53</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">8<span style="mso-tab-count: 1"> </span>4<span style="mso-tab-count: 1"> </span>61<span style="mso-tab-count: 1"> </span>
14, 16, 18, 48</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">9<span style="mso-tab-count: 1"> </span>3<span style="mso-tab-count: 1"> </span>19<span style="mso-tab-count: 1"> </span>
10, 13, 14</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">10<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>11</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">11<span style="mso-tab-count: 1"> </span>?<span style="mso-tab-count: 1"> </span>39916801<span style="mso-tab-count: 1">
</span>46, 77, 85 [no more through B = 500]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">12<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>13</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">13<span style="mso-tab-count: 1"> </span>5<span style="mso-tab-count: 1"> </span>83<span style="mso-tab-count: 1"> </span>
21, 26, 29, 44, 45</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">14<span style="mso-tab-count: 1"> </span>4<span style="mso-tab-count: 1"> </span>23<span style="mso-tab-count: 1"> </span>
16, 17, 18, 22</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">15<span style="mso-tab-count: 1"> </span>4<span style="mso-tab-count: 1"> </span>59<span style="mso-tab-count: 1"> </span>
19, 20, 21, 29</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">16<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>17</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">17<span style="mso-tab-count: 1"> </span>3+<span style="mso-tab-count: 1"> </span>661<span style="mso-tab-count: 1"> </span>
46, 183, 560</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">18<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>19</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">19<span style="mso-tab-count: 1"> </span>6<span style="mso-tab-count: 1"> </span>73<span style="mso-tab-count: 1"> </span>
23, 26, 38, 42, 45, 50</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">20<span style="mso-tab-count: 1"> </span>5+<span style="mso-tab-count: 1"> </span>20639383<span style="mso-tab-count: 1">
</span>22, 24, 29, 32, 130, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">21<span style="mso-tab-count: 1"> </span>4<span style="mso-tab-count: 1"> </span>43<span style="mso-tab-count: 1"> </span>
24, 32, 36, 39</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">22<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>23</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">23<span style="mso-tab-count: 1"> </span>5<span style="mso-tab-count: 1"> </span>47<span style="mso-tab-count: 1"> </span>
26, 33, 34, 35, 43</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">24<span style="mso-tab-count: 1"> </span>2+<span style="mso-tab-count: 1"> </span>811<span style="mso-tab-count: 1"> </span>
26, 90, … </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">25<span style="mso-tab-count: 1"> </span>2+<span style="mso-tab-count: 1"> </span>401<span style="mso-tab-count: 1"> </span>
31, 83, … </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">26<span style="mso-tab-count: 1"> </span>3+<span style="mso-tab-count: 1"> </span>1697<span style="mso-tab-count: 1"> </span>
29, 36, 89, … [no more through 90]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">27<span style="mso-tab-count: 1"> </span>2+<span style="mso-tab-count: 1"> </span>10888869450418352160768000001<span style="mso-tab-count: 2">
</span>56, 61, … </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">28<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>29</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">29<span style="mso-tab-count: 1"> </span>5+<span style="mso-tab-count: 1"> </span>14557<span style="mso-tab-count: 2">
</span>33, 37, 50, 62, 99, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">30<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>31</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">31<span style="mso-tab-count: 1"> </span>2+<span style="mso-tab-count: 1"> </span>257<span style="mso-tab-count: 1"> </span>
37, 87, … </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">32<span style="mso-tab-count: 1"> </span>2+<span style="mso-tab-count: 1"> </span>2281<span style="mso-tab-count: 1"> </span>
44, 97, … </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">33<span style="mso-tab-count: 1"> </span>1<span style="mso-tab-count: 1"> </span>67<span style="mso-tab-count: 1"> </span>
39</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">34<span style="mso-tab-count: 1"> </span>1+<span style="mso-tab-count: 1"> </span>67411<span style="mso-tab-count: 1"> </span>46 , …
</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">35<span style="mso-tab-count: 1"> </span>1+<span style="mso-tab-count: 1"> </span>137<span style="mso-tab-count: 1"> </span>
76, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">36<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>37</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">37<span style="mso-tab-count: 1"> </span>1+<span style="mso-tab-count: 1"> </span>13763753091226345046315979581580902400000001<span style="mso-tab-count: 1">
</span>90, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">38<span style="mso-tab-count: 1"> </span>1+<span style="mso-tab-count: 1"> </span>14029308060317546154181<span style="mso-tab-count: 2">
</span>62, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">39<span style="mso-tab-count: 1"> </span>1<span style="mso-tab-count: 1"> </span>79<span style="mso-tab-count: 1"> </span>
70</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">40<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>41</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">41<span style="mso-tab-count: 1"> </span>1+<span style="mso-tab-count: 1"> </span>33452526613163807108170062053440751665152000000001
<span style="mso-tab-count: 1"> </span>52, … </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">42<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>43</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">43<span style="mso-tab-count: 1"> </span>1<span style="mso-tab-count: 1"> </span>59<span style="mso-tab-count: 1"> </span>
56</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">44<span style="mso-tab-count: 1"> </span>1+<span style="mso-tab-count: 1"> </span>1694763<span style="mso-tab-count: 1">
</span>87, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">45<span style="mso-tab-count: 1"> </span>1+<span style="mso-tab-count: 1"> </span>293<span style="mso-tab-count: 1"> </span>
52, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">46<span style="mso-tab-count: 1"> </span>0<span style="mso-tab-count: 1"> </span>47</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">47<span style="mso-tab-count: 1"> </span>2+<span style="mso-tab-count: 1"> </span>6007711<span style="mso-tab-count: 1">
</span>50, 86, … </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">48<span style="mso-tab-count: 1"> </span>2+<span style="mso-tab-count: 1"> </span>12893<span style="mso-tab-count: 2">
</span>62, 68, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">49<span style="mso-tab-count: 1"> </span>2+<span style="mso-tab-count: 1"> </span>1021<span style="mso-tab-count: 2">
</span>76, 83, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">50<span style="mso-tab-count: 1"> </span>1+<span style="mso-tab-count: 1"> </span>149<span style="mso-tab-count: 2">
</span>111, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">-------------------------------------------------------------------</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">We note the sequence of record values of smallest B as a function of increasing A:</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">{4, 8, 14, 42, 46, 56, 76, 87, 111.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Meaning:</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=3, min B = 4;</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=7, min B = 8;</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=8, min B = 14;</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=11, min B = 42;</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=17, min B = 46;</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=27, min B = 56;</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=35, min B = 76;</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=44, min B = 87;</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">A=50, min B = 111.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">There are other interesting integer sequences implicit in the major table.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Further investigations will be detailed in the future.</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"> </p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Sum3Factorials.doc</font></p><br><br>
<div><span class="gmail_quote">On 8/14/06, <b class="gmail_sendername"><a href="mailto:franktaw@netscape.net">franktaw@netscape.net</a></b> <<a href="mailto:franktaw@netscape.net">franktaw@netscape.net</a>> wrote:</span>
<blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0px 0px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">Sequence (1) is problematic, in that each row is finite. If you fill<br>with zeros, the table winds up being mostly zeros. Better, I think, to
<br>just put the solutions for a given n into the same row. Then you can<br>advertise sequence (4) as the row lengths. You'll have to put the<br>later values into an extension, since you don't know all the primes for<br>
some relatively small n.<br><br>Franklin T. Adams-Watters<br><br><br>-----Original Message-----<br>From: Jonathan Post <<a href="mailto:jvospost3@gmail.com">jvospost3@gmail.com</a>><br><br>I agree, Hugo. I was thinking of 4 sequences:
<br>(1) The array of n and k such that n! + k! + 1 = a prime, n<k, read by<br>antidiagonals;<br>(2) a(n) = the least k such that n! + k! + 1 = a prime, or 0 if none;<br>(3) a(n) = the greatest k such that n! + k! + 1 = a prime, or 0 if none;
<br>(4) a(n) the number of solutions, i.e. the number of k such that n! +<br>k! + 1 = a prime, from 0 to upper bound A051301(n).<br><br>Or is there a way to compress this into fewer sequences?<br><br>I'm not saying that it's inherently interesting, but that it builds on
<br>existing OEIS seqs such as A002981, A088332, A089359, is elementary, is<br>new, is not trivial, allows some number theory to be taught to<br>non-science-major Math students such as I had for many semesters.<br><br>I know quite a few 1 + A! + B! + C! primes but those don't compress
<br>into a few sequences, 3-D arrays not as easy as 2-D.<br><br><br><br>On 8/14/06, Hugo Pfoertner <<a href="mailto:all@abouthugo.de">all@abouthugo.de</a>> wrote:Joshua Zucker wrote:<br><br>[...]<br><br>> So someone
<br>> should still check 693 and 845 ...<br><br>PFGW Version 1.2.0 for Windows [FFT v23.8]<br>Primality testing 693!+11!+1 [N-1, Brillhart-Lehmer-Selfridge]<br>Running N-1 test using base 13<br>693!+11!+1 is PRP! (69.7657s+0.0193s
)<br>Done.<br><br>PFGW Version 1.2.0 for Windows [FFT v23.8]<br>Primality testing 845!+11!+1 [N-1, Brillhart-Lehmer-Selfridge]<br>Running N-1 test using base 13<br>Running N-1 test using base 29<br>845!+11!+1 is PRP! (237.4787s+0.0485s
)<br>Done.<br><br>The given times are on a 800 Mhz Athlon, running another job. I am<br>currently preparing tests for the next round of Al Zimmermann's<br>programming contest, which will be on self-avoiding zigzag paths<br>
("matchstick snakes")<br><br>><br>[...]<br>><br>> For 1! + 17! + n!, my system produced the following candidate primes:<br>> 11 14 46 183 560<br><br>Confirmed with pfgw:<br>11!+1+17! is 3-PRP! (0.1049s+0.9518s
)<br>14!+1+17! is 3-PRP! (0.0001s+0.0538s)<br>46!+1+17! is 3-PRP! (0.0003s+0.0415s)<br>Switching to Exponentiating using Woltman FFT's<br>183!+1+17! is 3-PRP! (0.9605s+0.9951s)<br>560!+1+17! is 3-PRP! (30.1122s+6.2870s)<br>
<br>checked:<br>PFGW Version 1.2.0 for Windows [FFT v23.8]<br>Primality testing 560!+17!+1 [N-1, Brillhart-Lehmer-Selfridge]<br>Running N-1 test using base 19<br>Running N-1 test using base 23<br>560!+17!+1 is PRP! (75.0765s+0.0597s
)<br>Done.<br>Primality testing 183!+17!+1 [N-1, Brillhart-Lehmer-Selfridge]<br>Running N-1 test using base 19<br>Running N-1 test using base 23<br>Running N-1 test using base 37<br>183!+17!+1 is PRP! (9.1043s+0.0516s)<br>
Done.<br>><br>[...]<br>> --Joshua Zucker<br><br>To JVP:<br>Please don't conclude from my contribution that I find this problem very<br>interesting; I just wanted to confirm Joshua's results.<br><br>What I definitely don't support is a series of new sequences:
<br><br>Numbers n such that n!+k!+1 is prime, for k=1...100.<br><br>Hugo Pfoertner<br><br><br><br><br></blockquote></div><br>