<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>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">10-11 Aug 2006</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">These early results have been checked, but may still have errors. The author would be grateful if these were pointed out.</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">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 factorials can be prime is if the sum is of the form n! + 1, the so-called "factorial primes"
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<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)
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<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:
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<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>
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<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">We now tabularize and partial characterize Prime Sums of Exactly Three Distinct Factorials.</font></p>
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<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>
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<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 after which all higher A gives the same common prime factor to A! + B! + 1!, and it is not clear under what circumstances "several" means infinite.
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<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">(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">(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">(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"> </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, …}.</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>
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<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, uses the convention that "0" means either that there is no solution at all for the given A, and/or after the greatest given value in the row there are no greater solutions.
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<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 \ 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">1<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">2<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">3<span style="mso-tab-count: 1"> </span>4, 5, 6, 0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">4<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">5<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">6<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">7<span style="mso-tab-count: 1"> </span>8, 12, 16, 23, 27, 33, 37, 42, 53, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">8<span style="mso-tab-count: 1"> </span>14, 16, 18, 48, … [no more through 60]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">9<span style="mso-tab-count: 1"> </span>10, 13, 14, 0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">10<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">11<span style="mso-tab-count: 1"> </span>42,… [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">12<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">13<span style="mso-tab-count: 1"> </span>21, 26, 29, 44, 45, … [no more through 60]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">14<span style="mso-tab-count: 1"> </span>16, 17, 18, 22, 0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">15<span style="mso-tab-count: 1"> </span>19, 20, 21, 29, 0<span style="mso-spacerun: yes"> </span>[eventual common factor 59]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">16<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">17<span style="mso-tab-count: 1"> </span>46, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">18<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">19<span style="mso-tab-count: 1"> </span>23, 26, 38, 42, 45, 50, 0 [eventual common factor 73]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">20<span style="mso-tab-count: 1"> </span>22, 24, 29, 32, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">21<span style="mso-tab-count: 1"> </span>24, 32, 36, 39, 0 [eventual common factor 43]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">22<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">23<span style="mso-tab-count: 1"> </span>26, 33, 34, 35, 43, 0 [eventual common factor 47]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">24<span style="mso-tab-count: 1"> </span>26, … [no more through 60]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">25<span style="mso-tab-count: 1"> </span>31, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">26<span style="mso-tab-count: 1"> </span>29, 36, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">27<span style="mso-tab-count: 1"> </span>56, 61, … [no more through 80]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">28<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">29<span style="mso-tab-count: 1"> </span>33, 37, 50, 62, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">30<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">31<span style="mso-tab-count: 1"> </span>37, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">32<span style="mso-tab-count: 1"> </span>44, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">33<span style="mso-tab-count: 1"> </span>39, 0 [eventual common factor 67]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">34<span style="mso-tab-count: 1"> </span>46 , … [no more through 80]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">35<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</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">37<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>62, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">39<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</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">41<span style="mso-tab-count: 1"> </span>52, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">42<span style="mso-tab-count: 1"> </span>0</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">43<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>87, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">45<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</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">47<span style="mso-tab-count: 1"> </span>50, … [no more through 70]</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">48<span style="mso-tab-count: 1"> </span>62, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">49<span style="mso-tab-count: 1"> </span>76, …</font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">50<span style="mso-tab-count: 1"> </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"><font face="Times New Roman"> </font></p>
<p class="MsoNormal" style="MARGIN: 0in 0in 0pt"><font face="Times New Roman">Sum3Factorials.doc</font></p>