Is ${F_{n}}^2 - 28$ always a composite number?
${F_n}^2 - 28$ indeed can not be a prime number for $n>5$.
Let's suppose that ${F_n}^2 - 28$ is a prime. In that case, $n$ must be even, this was shown (easily using congruences) in other answers, so I'll omit this part for space saving. I'll also use known identity $5{F_n}^2 = {L_n}^2 - 4$ valid for even $n$ ($L_n$ is $n$-th Lucas number). It follows that that:
$$5({F_n}^2-28) = 5{F_n}^2-140 = ({L_n}^2 - 4) - 140 = {L_n}^2-144 = (L_n-12)(L_n+12)$$
${F_n}^2 - 28$ could be prime only if one of $L_n - 12$ and $L_n + 12$ are $1$, $5$, $-1$, $-5$, and those are just few cases of small $n$. For all others, ${F_n}^2 - 28$ can not be a prime.
Possible generalization: Expressions ${F_n}^2 - (k^2 - 4)/5$ (k is any natural number) could be candidates for a similar proof. However, some other bits and peaces (like proving that $n$ must be even) must be valid too, and this is often not true. However, it looks that, for instance, ${F_n}^2 - 145$ satisfies all conditions, and also can be reduced to $({L_n}^2 - 4) - 725$ and finally to $(L_n-27)(L_n+27)$.
Outside such generalization, it seems that the original statement (or its slightly modified version related only to condition $n>5$) is valid for ${F_n}^2 - 13$, ${F_n}^2 - 31$, ${F_n}^2 - 45$, ${F_n}^2 - 58$, ${F_n}^2 - 78$, ${F_n}^2 - 85$, ${F_n}^2 - 91$, ${F_n}^2 - 115$, ${F_n}^2 - 133$, ${F_n}^2 - 142$, ${F_n}^2 - 154$, ${F_n}^2 - 175$, ${F_n}^2 - 211$, ${F_n}^2 - 217$. (warning: I checked then with the help of a computer, and only for $n<1000$; please see comments for an interesting example involving $n>1000$)
Now, for the sake of curiosity, let us take a look at one of these cases. Case ${F_n}^2 - 85$ can be proven using congruences for $2$, $3$, $17$, and $107$ only. This means that one of $2$, $3$, $17$, and $107$ is always a divisor of ${F_n}^2 - 85$. Original case ${F_n}^2 - 28$ is fundamentally different, and, if I may say, "more beautiful", in the sense that it can have (and often has) only huge divisors, like in these examples:
$${F_{20}}^2 - 28 = 3023\times15139$$ $${F_{172}}^2 - 28 = 10348333\times53505724471\times315619755257\times10455376853041\times82024860865049\times1040059595540327$$
For users of Mathematica, this code:
Select[Table[Fibonacci[n], {n, 1, 1000}], PrimeQ[#*# + 3] &]
returns primes of the form ${F_n}^2+3$, for $1\le n\le 1000$. In this case there are $6$ such primes:
{
2, 8, 3524578, 27777890035288, 2011595611835338993891308327102733537615455242513357158345612749706882\ 9146295425939723629305572732574726246290673965789878845363842331040064\ 16432124798818261534841714338, 2949592466076064248964701302014885591673737506156850406413751530665307\ 5810241060939483954895520932111023343610904846943097162533007651451709\ 723277579925520157875345780869307228929160}
You're correct: $F_n^2-28$ is never a prime for $n\geq 6$, i.e., all $n$ where $F_n^2-28\gt 0$; arguably it's prime for $n=4$ ($F_4^2-28=-19$) and $n=5$ ($F_5^2-28=-3$).
First of all, note that if $3\not\mid F_n$, then $F_n^2\equiv 1\equiv 28\pmod 3$, so $3\mid F_n^2-28$. We can therefore assume in what follows that $3\mid F_n$ and therefore that $4\mid n$.
Now, (defining $G_n=F_n^2-28$ for convenience in what follows:) numeric evidence suggests the conjecture that for all $n$ divisible by $4$ there's some $a$ with $G_n=5a^2+24a = a\times(5a+24)$. We can prove this (and thus provide a factorization for all $n$) as follows:
Suppose that $G_n=5a^2+24a$; in other words, $5a^2+24a-G_n=0$. The (positive) solution to this quadratic equation is $a=\frac1{10}\left(-24+\sqrt{24^2+20G_n}\right)$. We'll first show that this quantity is rational, and then that it's integral. Note that $24^2+20G_n=24^2+20(F_n^2-28)=16+20F_n^2$, so $\sqrt{24^2+20G_n}=2\sqrt{4+5F_n^2}$. But since $n$ is even, we have $4+5F_n^2=L_n^2$ where $L_n$ is the $n$th Lucas number (a sort of conjugate to the Fibonacci numbers, satisfying the same recurrence). Clearly $2L_n-24$ is divisible by $2$; its divisibility by $5$ is equivalent to saying that $2L_n\equiv -1\bmod 5$ or that $L_n=2\bmod 5$. But this follows since, as noted at the start, we're in the case $4\mid n$ (and because the period of the Lucas numbers mod $5$ is $4$). This implies the integrality of $a$, which in turn implies the desired factorization of $G_n$.
The recurrence for the squares of the Fibonacci numbers is $$ x_n=2x_{n-1}+2x_{n-2}-x_{n-3} $$ Thus a repeat of a subsequence of length $3$ implies a repeat of the entire sequence.
mod $2$, we get $$ \overbrace{0,1,1},\overbrace{0,1,1},\dots $$ Thus, for $n\equiv0\pmod3$, we have that $F_n^2-28\equiv0\pmod2$
mod $3$, we get $$ \overbrace{0,1,1},1,\overbrace{0,1,1},\dots $$ Thus, for $n\not\equiv0\pmod4$, we have $F_n^2-28\equiv0\pmod3$
mod $7$, we get $$ \overbrace{0,1,1},4,2,4,1,1,\overbrace{0,1,1},\dots $$ Thus, for $n\equiv0\pmod8$, we have that $F_n^2-28\equiv0\pmod7$
The only thing not covered above is $n\equiv4\pmod{24}$ and $n\equiv20\pmod{24}$.
Recall that $F_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ where $\alpha$ and $\beta$ are the roots of $x^2-x-1=0$. Note that $(\alpha-\beta)^2=5$ and $\alpha\beta=-1$. Using these relations, it is not difficult to obtain $$ \begin{align} 5(F_{2k}^2-28) &=5F_{2k}^2-140\\ &=\alpha^{4k}+\beta^{4k}-142\\ &=\left(\alpha^{2k}+\beta^{2k}\right)^2-144\\ &=\left(L_{2k}-12\right)\left(L_{2k}+12\right) \end{align} $$ where $L_n=\alpha^n+\beta^n$ is a Lucas Number. In an argument similar to those above, it can be shown that $L_{2k}-12\equiv0\pmod5$ for even $k$ and $L_{2k}+12\equiv0\pmod5$ for odd $k$.
This covers the case of all even values of $n$ for $n\ge6$ since $L_6=18$ and $L_n$ is monotonically increasing after that.
The cases above cover all $n\ge6$, which are the cases for which $F_n^2-28\ge0$. Specifically:
If $n$ is odd, $F_n^2-28=3\cdot\frac{F_n^2-28}3$
If $n=0\pmod4$, $F_n^2-28=\frac{L_n-12}{5}(L_n+12)$
If $n=2\pmod4$, $F_n^2-28=9\cdot\frac{L_n-12}3\frac{L_n+12}{15}$
The argument above can be simplified by ignoring the cases mod $2$ and mod $7$ since only the case mod $3$ and the Lucas number argument is needed.
I see that the Lucas number identity has already been noted in another answer while I've worked on this answer, but I will include it for completeness.
After Steven and robjohn's answers, it seemed useful to write a command (C++ with GMP) to do Fermat's factoring, try to write some large number $n$ as $x^2 - y^2$ with small $y.$ slight generalization, needed for this problem, write $$ \color{magenta}{ zn = x^2 - y^2} $$ with small $z,y.$ So, I did that; much variety for the first few. See how this method, stopped early, does not detect small factors such as $2,3,7.$ If there are several ways to do the task, the program prints out the first dozen $(z,y)$ pairs; in any case, $z^2 + y^2$ is no larger than a bound provided by the programmer (me).
6 8 Fermat: ( 1 , 0 ); ( 1 , 8 ); ( 2 , 3 ); ( 2 , 7 ); ( 2 , 17 ); ( 3 , 6 ); ( 3 , 26 ); ( 4 , 0 ); ( 4 , 5 ); ( 4 , 9 ); ( 4 , 16 ); ( 5 , 4 );
7 13 Fermat: ( 1 , 22 ); ( 3 , 19 ); ( 5 , 16 ); ( 7 , 13 ); ( 9 , 10 ); ( 11 , 7 ); ( 13 , 4 ); ( 15 , 1 ); ( 17 , 2 ); ( 19 , 5 ); ( 21 , 8 ); ( 23 , 11 );
8 21 Fermat: ( 1 , 26 ); ( 3 , 19 ); ( 5 , 12 ); ( 7 , 5 ); ( 9 , 2 ); ( 11 , 9 ); ( 13 , 16 ); ( 15 , 23 ); ( 20 , 24 ); ( 24 , 17 ); ( 28 , 10 );
9 34 Fermat: ( 3 , 29 ); ( 4 , 23 ); ( 5 , 17 ); ( 6 , 11 ); ( 7 , 5 ); ( 8 , 1 ); ( 9 , 7 ); ( 10 , 13 ); ( 11 , 19 ); ( 12 , 25 ); ( 25 , 19 ); ( 26 , 16 );
10 55 Fermat: ( 1 , 22 ); ( 3 , 15 ); ( 5 , 12 ); ( 8 , 7 ); ( 16 , 3 ); ( 20 , 24 ); ( 21 , 8 );
11 89 Fermat:
12 144 Fermat: ( 5 , 12 ); ( 6 , 19 ); ( 20 , 24 ); ( 22 , 7 );
13 233 Fermat:
14 377 Fermat: ( 5 , 12 ); ( 20 , 24 );
15 610 Fermat: ( 2 , 25 ); ( 17 , 1 );
16 987 Fermat: ( 5 , 12 ); ( 20 , 24 );
17 1597 Fermat:
18 2584 Fermat: ( 5 , 12 ); ( 20 , 24 );
19 4181 Fermat:
20 6765 Fermat: ( 5 , 12 ); ( 20 , 24 );
21 10946 Fermat:
22 17711 Fermat: ( 5 , 12 ); ( 20 , 24 );
23 28657 Fermat:
24 46368 Fermat: ( 5 , 12 ); ( 20 , 24 );
25 75025 Fermat:
26 121393 Fermat: ( 5 , 12 ); ( 20 , 24 );
27 196418 Fermat:
28 317811 Fermat: ( 5 , 12 ); ( 20 , 24 );
29 514229 Fermat:
30 832040 Fermat: ( 5 , 12 ); ( 20 , 24 );
31 1346269 Fermat:
32 2178309 Fermat: ( 5 , 12 ); ( 20 , 24 );
33 3524578 Fermat:
34 5702887 Fermat: ( 5 , 12 ); ( 20 , 24 );
35 9227465 Fermat:
36 14930352 Fermat: ( 5 , 12 ); ( 20 , 24 );
37 24157817 Fermat:
38 39088169 Fermat: ( 5 , 12 ); ( 20 , 24 );
39 63245986 Fermat:
40 102334155 Fermat: ( 5 , 12 ); ( 20 , 24 );
41 165580141 Fermat:
42 267914296 Fermat: ( 5 , 12 ); ( 20 , 24 );
43 433494437 Fermat:
44 701408733 Fermat: ( 5 , 12 ); ( 20 , 24 );
45 1134903170 Fermat:
46 1836311903 Fermat: ( 5 , 12 ); ( 20 , 24 );
47 2971215073 Fermat:
48 4807526976 Fermat: ( 5 , 12 ); ( 20 , 24 );
49 7778742049 Fermat:
50 12586269025 Fermat: ( 5 , 12 ); ( 20 , 24 );
51 20365011074 Fermat:
52 32951280099 Fermat: ( 5 , 12 ); ( 20 , 24 );
53 53316291173 Fermat:
54 86267571272 Fermat: ( 5 , 12 ); ( 20 , 24 );
55 139583862445 Fermat:
56 225851433717 Fermat: ( 5 , 12 ); ( 20 , 24 );
57 365435296162 Fermat:
58 591286729879 Fermat: ( 5 , 12 ); ( 20 , 24 );
59 956722026041 Fermat:
60 1548008755920 Fermat: ( 5 , 12 ); ( 20 , 24 );
61 2504730781961 Fermat:
62 4052739537881 Fermat: ( 5 , 12 ); ( 20 , 24 );
63 6557470319842 Fermat:
64 10610209857723 Fermat: ( 5 , 12 ); ( 20 , 24 );
65 17167680177565 Fermat:
66 27777890035288 Fermat: ( 5 , 12 ); ( 20 , 24 );
67 44945570212853 Fermat:
68 72723460248141 Fermat: ( 5 , 12 ); ( 20 , 24 );
69 117669030460994 Fermat:
70 190392490709135 Fermat: ( 5 , 12 ); ( 20 , 24 );
71 308061521170129 Fermat:
72 498454011879264 Fermat: ( 5 , 12 ); ( 20 , 24 );
73 806515533049393 Fermat:
74 1304969544928657 Fermat: ( 5 , 12 ); ( 20 , 24 );
75 2111485077978050 Fermat:
76 3416454622906707 Fermat: ( 5 , 12 ); ( 20 , 24 );
77 5527939700884757 Fermat:
78 8944394323791464 Fermat: ( 5 , 12 ); ( 20 , 24 );
79 14472334024676221 Fermat:
80 23416728348467685 Fermat: ( 5 , 12 ); ( 20 , 24 );
81 37889062373143906 Fermat:
82 61305790721611591 Fermat: ( 5 , 12 ); ( 20 , 24 );
83 99194853094755497 Fermat:
84 160500643816367088 Fermat: ( 5 , 12 ); ( 20 , 24 );
85 259695496911122585 Fermat:
86 420196140727489673 Fermat: ( 5 , 12 ); ( 20 , 24 );
87 679891637638612258 Fermat:
88 1100087778366101931 Fermat: ( 5 , 12 ); ( 20 , 24 );
89 1779979416004714189 Fermat:
90 2880067194370816120 Fermat: ( 5 , 12 ); ( 20 , 24 );
91 4660046610375530309 Fermat:
92 7540113804746346429 Fermat: ( 5 , 12 ); ( 20 , 24 );
93 12200160415121876738 Fermat:
94 19740274219868223167 Fermat: ( 5 , 12 ); ( 20 , 24 );
95 31940434634990099905 Fermat:
96 51680708854858323072 Fermat: ( 5 , 12 ); ( 20 , 24 );
97 83621143489848422977 Fermat:
98 135301852344706746049 Fermat: ( 5 , 12 ); ( 20 , 24 );
99 218922995834555169026 Fermat:
100 354224848179261915075 Fermat: ( 5 , 12 ); ( 20 , 24 );
101 573147844013817084101 Fermat:
102 927372692193078999176 Fermat: ( 5 , 12 ); ( 20 , 24 );
103 1500520536206896083277 Fermat:
104 2427893228399975082453 Fermat: ( 5 , 12 ); ( 20 , 24 );
105 3928413764606871165730 Fermat:
106 6356306993006846248183 Fermat: ( 5 , 12 ); ( 20 , 24 );
107 10284720757613717413913 Fermat:
108 16641027750620563662096 Fermat: ( 5 , 12 ); ( 20 , 24 );
109 26925748508234281076009 Fermat:
110 43566776258854844738105 Fermat: ( 5 , 12 ); ( 20 , 24 );
111 70492524767089125814114 Fermat:
112 114059301025943970552219 Fermat: ( 5 , 12 ); ( 20 , 24 );
113 184551825793033096366333 Fermat:
114 298611126818977066918552 Fermat: ( 5 , 12 ); ( 20 , 24 );
115 483162952612010163284885 Fermat:
116 781774079430987230203437 Fermat: ( 5 , 12 ); ( 20 , 24 );
117 1264937032042997393488322 Fermat:
118 2046711111473984623691759 Fermat: ( 5 , 12 ); ( 20 , 24 );
119 3311648143516982017180081 Fermat:
120 5358359254990966640871840 Fermat: ( 5 , 12 ); ( 20 , 24 );
121 8670007398507948658051921 Fermat:
122 14028366653498915298923761 Fermat: ( 5 , 12 ); ( 20 , 24 );
123 22698374052006863956975682 Fermat:
124 36726740705505779255899443 Fermat: ( 5 , 12 ); ( 20 , 24 );
125 59425114757512643212875125 Fermat:
126 96151855463018422468774568 Fermat: ( 5 , 12 ); ( 20 , 24 );
127 155576970220531065681649693 Fermat:
128 251728825683549488150424261 Fermat: ( 5 , 12 ); ( 20 , 24 );
129 407305795904080553832073954 Fermat:
130 659034621587630041982498215 Fermat: ( 5 , 12 ); ( 20 , 24 );
131 1066340417491710595814572169 Fermat:
132 1725375039079340637797070384 Fermat: ( 5 , 12 ); ( 20 , 24 );
133 2791715456571051233611642553 Fermat:
134 4517090495650391871408712937 Fermat: ( 5 , 12 ); ( 20 , 24 );
135 7308805952221443105020355490 Fermat:
136 11825896447871834976429068427 Fermat: ( 5 , 12 ); ( 20 , 24 );
137 19134702400093278081449423917 Fermat:
138 30960598847965113057878492344 Fermat: ( 5 , 12 ); ( 20 , 24 );
139 50095301248058391139327916261 Fermat:
140 81055900096023504197206408605 Fermat: ( 5 , 12 ); ( 20 , 24 );
141 131151201344081895336534324866 Fermat:
142 212207101440105399533740733471 Fermat: ( 5 , 12 ); ( 20 , 24 );
143 343358302784187294870275058337 Fermat:
144 555565404224292694404015791808 Fermat: ( 5 , 12 ); ( 20 , 24 );
145 898923707008479989274290850145 Fermat:
146 1454489111232772683678306641953 Fermat: ( 5 , 12 ); ( 20 , 24 );
147 2353412818241252672952597492098 Fermat:
148 3807901929474025356630904134051 Fermat: ( 5 , 12 ); ( 20 , 24 );
149 6161314747715278029583501626149 Fermat:
150 9969216677189303386214405760200 Fermat: ( 5 , 12 ); ( 20 , 24 );
151 16130531424904581415797907386349 Fermat:
152 26099748102093884802012313146549 Fermat: ( 5 , 12 ); ( 20 , 24 );
153 42230279526998466217810220532898 Fermat:
154 68330027629092351019822533679447 Fermat: ( 5 , 12 ); ( 20 , 24 );
155 110560307156090817237632754212345 Fermat:
156 178890334785183168257455287891792 Fermat: ( 5 , 12 ); ( 20 , 24 );
157 289450641941273985495088042104137 Fermat:
158 468340976726457153752543329995929 Fermat: ( 5 , 12 ); ( 20 , 24 );
159 757791618667731139247631372100066 Fermat:
160 1226132595394188293000174702095995 Fermat: ( 5 , 12 ); ( 20 , 24 );
161 1983924214061919432247806074196061 Fermat:
162 3210056809456107725247980776292056 Fermat: ( 5 , 12 ); ( 20 , 24 );
163 5193981023518027157495786850488117 Fermat:
164 8404037832974134882743767626780173 Fermat: ( 5 , 12 ); ( 20 , 24 );
165 13598018856492162040239554477268290 Fermat:
166 22002056689466296922983322104048463 Fermat: ( 5 , 12 ); ( 20 , 24 );
167 35600075545958458963222876581316753 Fermat:
168 57602132235424755886206198685365216 Fermat: ( 5 , 12 ); ( 20 , 24 );
169 93202207781383214849429075266681969 Fermat:
170 150804340016807970735635273952047185 Fermat: ( 5 , 12 ); ( 20 , 24 );
171 244006547798191185585064349218729154 Fermat:
172 394810887814999156320699623170776339 Fermat: ( 5 , 12 ); ( 20 , 24 );
173 638817435613190341905763972389505493 Fermat:
174 1033628323428189498226463595560281832 Fermat: ( 5 , 12 ); ( 20 , 24 );
175 1672445759041379840132227567949787325 Fermat:
176 2706074082469569338358691163510069157 Fermat: ( 5 , 12 ); ( 20 , 24 );
177 4378519841510949178490918731459856482 Fermat:
178 7084593923980518516849609894969925639 Fermat: ( 5 , 12 ); ( 20 , 24 );
179 11463113765491467695340528626429782121 Fermat:
180 18547707689471986212190138521399707760 Fermat: ( 5 , 12 ); ( 20 , 24 );
181 30010821454963453907530667147829489881 Fermat:
182 48558529144435440119720805669229197641 Fermat: ( 5 , 12 ); ( 20 , 24 );
183 78569350599398894027251472817058687522 Fermat:
184 127127879743834334146972278486287885163 Fermat: ( 5 , 12 ); ( 20 , 24 );
185 205697230343233228174223751303346572685 Fermat:
186 332825110087067562321196029789634457848 Fermat: ( 5 , 12 ); ( 20 , 24 );
187 538522340430300790495419781092981030533 Fermat:
188 871347450517368352816615810882615488381 Fermat: ( 5 , 12 ); ( 20 , 24 );
189 1409869790947669143312035591975596518914 Fermat:
190 2281217241465037496128651402858212007295 Fermat: ( 5 , 12 ); ( 20 , 24 );
191 3691087032412706639440686994833808526209 Fermat:
192 5972304273877744135569338397692020533504 Fermat: ( 5 , 12 ); ( 20 , 24 );
193 9663391306290450775010025392525829059713 Fermat:
194 15635695580168194910579363790217849593217 Fermat: ( 5 , 12 ); ( 20 , 24 );
195 25299086886458645685589389182743678652930 Fermat:
196 40934782466626840596168752972961528246147 Fermat: ( 5 , 12 ); ( 20 , 24 );
197 66233869353085486281758142155705206899077 Fermat:
198 107168651819712326877926895128666735145224 Fermat: ( 5 , 12 ); ( 20 , 24 );
199 173402521172797813159685037284371942044301 Fermat:
200 280571172992510140037611932413038677189525 Fermat: ( 5 , 12 ); ( 20 , 24 );
201 453973694165307953197296969697410619233826 Fermat:
202 734544867157818093234908902110449296423351 Fermat: ( 5 , 12 ); ( 20 , 24 );
203 1188518561323126046432205871807859915657177 Fermat:
204 1923063428480944139667114773918309212080528 Fermat: ( 5 , 12 ); ( 20 , 24 );
205 3111581989804070186099320645726169127737705 Fermat:
206 5034645418285014325766435419644478339818233 Fermat: ( 5 , 12 ); ( 20 , 24 );
207 8146227408089084511865756065370647467555938 Fermat:
208 13180872826374098837632191485015125807374171 Fermat: ( 5 , 12 ); ( 20 , 24 );
209 21327100234463183349497947550385773274930109 Fermat:
210 34507973060837282187130139035400899082304280 Fermat: ( 5 , 12 ); ( 20 , 24 );
211 55835073295300465536628086585786672357234389 Fermat:
212 90343046356137747723758225621187571439538669 Fermat: ( 5 , 12 ); ( 20 , 24 );
213 146178119651438213260386312206974243796773058 Fermat:
214 236521166007575960984144537828161815236311727 Fermat: ( 5 , 12 ); ( 20 , 24 );
215 382699285659014174244530850035136059033084785 Fermat:
216 619220451666590135228675387863297874269396512 Fermat: ( 5 , 12 ); ( 20 , 24 );
217 1001919737325604309473206237898433933302481297 Fermat:
218 1621140188992194444701881625761731807571877809 Fermat: ( 5 , 12 ); ( 20 , 24 );
219 2623059926317798754175087863660165740874359106 Fermat:
220 4244200115309993198876969489421897548446236915 Fermat: ( 5 , 12 ); ( 20 , 24 );
221 6867260041627791953052057353082063289320596021 Fermat:
222 11111460156937785151929026842503960837766832936 Fermat: ( 5 , 12 ); ( 20 , 24 );
223 17978720198565577104981084195586024127087428957 Fermat:
224 29090180355503362256910111038089984964854261893 Fermat: ( 5 , 12 ); ( 20 , 24 );
225 47068900554068939361891195233676009091941690850 Fermat:
226 76159080909572301618801306271765994056795952743 Fermat: ( 5 , 12 ); ( 20 , 24 );
227 123227981463641240980692501505442003148737643593 Fermat:
228 199387062373213542599493807777207997205533596336 Fermat: ( 5 , 12 ); ( 20 , 24 );
229 322615043836854783580186309282650000354271239929 Fermat:
230 522002106210068326179680117059857997559804836265 Fermat: ( 5 , 12 ); ( 20 , 24 );
231 844617150046923109759866426342507997914076076194 Fermat:
232 1366619256256991435939546543402365995473880912459 Fermat: ( 5 , 12 ); ( 20 , 24 );
233 2211236406303914545699412969744873993387956988653 Fermat:
234 3577855662560905981638959513147239988861837901112 Fermat: ( 5 , 12 ); ( 20 , 24 );
jagy@phobeusjunior:
Here is the command:
string mp_Factored_Fermat( mpz_class i, int bound)
{
int squarefac = 0;
string fac;
fac = " Fermat: ";
int count = 0;
for(int z = 1; count < 12 && z * z <= bound; ++z){
for(int x = 0; count < 12 && x * x + z * z <= bound; ++x){
if ( mp_SquareQ( z * i + x * x ) )
{
++count;
fac += " ( ";
fac += stringify( z) ;
fac += " , ";
fac += stringify( x) ;
fac += " ); ";
}
}}
return fac;
} // mp_Factored_Fermat
In case anyone gets interested, this calls
string stringify(unsigned int x)
{
ostringstream o;
o << x ;
return o.str();
}
I should find out whether there is a direct "stringify" command that makes a string from an mpz_class. Probably. YES. If n is an mpz_class, we get a C++ string from n.get_str()
Well, live and loin. The odd index entries are divisible by small primes, 2 or 3, detectable quickly by trial division, the even index entries have huge factors, detectable quickly by a slight modification of Fermat's favorite technique.
6 8 = 2^2 cdot 3^2
7 13 = 3 cdot 47
8 21 = 7 cdot 59
9 34 = 2^3 cdot 3 cdot 47
10 55 = 3^4 cdot 37
11 89 = 3^2 cdot 877
12 144 = 2^2 cdot 31 cdot 167
13 233 = 3^2 cdot 6029
14 377 = 3^3 cdot 19 cdot 277
15 610 = 2^3 cdot 3 cdot 37 cdot 419
16 987 = 7 cdot 317 cdot 439
17 1597 = 3 cdot 271 cdot 3137
18 2584 = 2^2 cdot 3^2 cdot 31^2 cdot 193
19 4181 = 3 cdot 5826911
20 6765 = 3023 cdot 15139
21 10946 = 2^3 cdot 3 cdot 4992287
22 17711 = 3^3 cdot 19 cdot 53 cdot 83 cdot 139
23 28657 = 3^2 cdot 37 cdot 47 cdot 137 cdot 383
24 46368 = 2^2 cdot 7 cdot 139 cdot 373 cdot 1481
25 75025 = 3^2 cdot 47 cdot 953 cdot 13963
26 121393 = 3^6 cdot 1117 cdot 18097
27 196418 = 2^3 cdot 3 cdot 6803 cdot 236293
28 317811 = 37 cdot 311 cdot 457 cdot 19207
29 514229 = 3 cdot 103 cdot 227 cdot 3769891
30 832040 = 2^2 cdot 3^2 cdot 62017 cdot 310081
31 1346269 = 3 cdot 604146740111
32 2178309 = 7 cdot 19 cdot 53 cdot 691 cdot 974167
33 3524578 = 2^3 cdot 3 cdot 517610419919
34 5702887 = 3^3 cdot 251 cdot 1093 cdot 1129 cdot 3889
35 9227465 = 3^4 cdot 197 cdot 223 cdot 23928127
36 14930352 = 2^2 cdot 727 cdot 22961 cdot 3338527
37 24157817 = 3^3 cdot 21614819340943
38 39088169 = 3^3 cdot 1942307 cdot 29134597
39 63245986 = 2^3 cdot 3 cdot 47 cdot 384487 cdot 9223063
40 102334155 = 7 cdot 19 cdot 103 cdot 449 cdot 14561 cdot 116927
41 165580141 = 3 cdot 47 cdot mbox{BIG}
42 267914296 = 2^2 cdot 3^2 cdot 31 cdot 587 cdot 34019 cdot 3220831
43 433494437 = 3 cdot 643 cdot 4575983 cdot 21288763
44 701408733 = 271 cdot 1157489 cdot 1568397619
45 1134903170 = 2^3 cdot 3 cdot 3391 cdot 26459 cdot 598143187
46 1836311903 = 3^4 cdot 1979 cdot 138323 cdot 152078453
47 2971215073 = 3^2 cdot 21821 cdot 44952207150209
48 4807526976 = 2^2 cdot 7 cdot 31 cdot 37 cdot 2099 cdot 9887 cdot 34677281
49 7778742049 = 3^2 cdot 69191 cdot 97168751659867
50 12586269025 = 3^3 cdot 19^2 cdot 59 cdot 137 cdot 643 cdot 3127083679
51 20365011074 = 2^3 cdot 3 cdot 3547 cdot 3947 cdot 1234325623303
52 32951280099 = 131 cdot 26431 cdot 557537 cdot 562452689
53 53316291173 = 3 cdot 37 cdot 10771 cdot mbox{BIG}
54 86267571272 = 2^2 cdot 3^2 cdot 141511 cdot 227191 cdot 6430005121
55 139583862445 = 3 cdot 47 cdot 1783 cdot 2939579 cdot 26364214181
56 225851433717 = 7 cdot 1487 cdot 5431 cdot 2656807 cdot 339622837
57 365435296162 = 2^3 cdot 3 cdot 47 cdot 19433 cdot 477767 cdot 12751341727
58 591286729879 = 3^3 cdot 19 cdot 29173 cdot mbox{BIG}
59 956722026041 = 3^2 cdot 19853 cdot mbox{BIG}
60 1548008755920 = 2^2 cdot 3089 cdot 116819 cdot mbox{BIG}
61 2504730781961 = 3^2 cdot 37 cdot 277 cdot 176317 cdot mbox{BIG}
62 4052739537881 = 3^4 cdot 83 cdot 20627 cdot 20693 cdot 39209 cdot 145978529
63 6557470319842 = 2^3 cdot 3 cdot 131 cdot mbox{BIG}
64 10610209857723 = 7 cdot 103 cdot 3407 cdot mbox{BIG}
65 17167680177565 = 3 cdot 2297 cdot mbox{BIG}
66 27777890035288 = 2^2 cdot 3^2 cdot 37 cdot 59 cdot 88169 cdot 398011 cdot 279789416173
67 44945570212853 = 3 cdot 131 cdot 1571 cdot 1901 cdot mbox{BIG}
68 72723460248141 = 19 cdot 139 cdot mbox{BIG}
69 117669030460994 = 2^3 cdot 3 cdot 20897 cdot mbox{BIG}
70 190392490709135 = 3^3 cdot 139 cdot 4451 cdot mbox{BIG}
71 308061521170129 = 3^3 cdot 47 cdot 1229 cdot mbox{BIG}
72 498454011879264 = 2^2 cdot 7 cdot 31 cdot 383 cdot 36877 cdot mbox{BIG}
73 806515533049393 = 3^4 cdot 47 cdot mbox{BIG}
74 1304969544928657 = 3^3 cdot 569 cdot 1627 cdot mbox{BIG}
75 2111485077978050 = 2^3 cdot 3 cdot 103 cdot 15091 cdot mbox{BIG}
76 3416454622906707 = 19 cdot 53 cdot 3838231 cdot mbox{BIG}
77 5527939700884757 = 3 cdot 1091 cdot 7109 cdot mbox{BIG}
78 8944394323791464 = 2^2 cdot 3^2 cdot 31 cdot 131 cdot mbox{BIG}
79 14472334024676221 = 3 cdot 203617 cdot mbox{BIG}
80 23416728348467685 = 7 cdot 2137 cdot 119087 cdot mbox{BIG}
81 37889062373143906 = 2^3 cdot 3 cdot 953 cdot 2801 cdot 2290829 cdot mbox{BIG}
82 61305790721611591 = 3^5 cdot 227 cdot 3929 cdot 10099 cdot 37579 cdot 1334947 cdot 34229552551
83 99194853094755497 = 3^2 cdot 168143 cdot mbox{BIG}
84 160500643816367088 = 2^2 cdot 68099 cdot 95063 cdot mbox{BIG}
85 259695496911122585 = 3^2 cdot 317 cdot mbox{BIG}
86 420196140727489673 = 3^3 cdot 19 cdot 37 cdot 53 cdot 4202911 cdot mbox{BIG}
87 679891637638612258 = 2^3 cdot 3 cdot 47 cdot 29401 cdot 357197 cdot mbox{BIG}
88 1100087778366101931 = 7 cdot 137 cdot 4903 cdot 10223 cdot 187373 cdot 10234897 cdot 13128204456583
89 1779979416004714189 = 3 cdot 47 cdot 1117 cdot mbox{BIG}
90 2880067194370816120 = 2^2 cdot 3^2 cdot 283 cdot 35363 cdot 5160157 cdot 5881991 cdot mbox{BIG}
91 4660046610375530309 = 3 cdot 37 cdot 811 cdot mbox{BIG}
92 7540113804746346429 = 197 cdot mbox{BIG}
93 12200160415121876738 = 2^3 cdot 3 cdot mbox{BIG}
94 19740274219868223167 = 3^3 cdot 19 cdot 7673 cdot 62851 cdot 137707 cdot mbox{BIG}
95 31940434634990099905 = 3^2 cdot 283 cdot 10369 cdot 1328077 cdot mbox{BIG}
96 51680708854858323072 = 2^2 cdot 7 cdot 223 cdot 6111047 cdot mbox{BIG}
97 83621143489848422977 = 3^2 cdot 59951 cdot mbox{BIG}
98 135301852344706746049 = 3^4 cdot 3491 cdot 7027 cdot mbox{BIG}
99 218922995834555169026 = 2^3 cdot 3 cdot 37 cdot 3527 cdot 402137 cdot mbox{BIG}
100 354224848179261915075 = 367 cdot 1063 cdot 860113 cdot mbox{BIG}
101 573147844013817084101 = 3 cdot 647 cdot 81703 cdot mbox{BIG}
102 927372692193078999176 = 2^2 cdot 3^2 cdot 31 cdot 617 cdot 1201 cdot 5813323 cdot mbox{BIG}
103 1500520536206896083277 = 3 cdot 47 cdot mbox{BIG}
104 2427893228399975082453 = 7 cdot 19 cdot 37 cdot 1121453 cdot mbox{BIG}
105 3928413764606871165730 = 2^3 cdot 3 cdot 47 cdot mbox{BIG}
106 6356306993006846248183 = 3^3 cdot 83 cdot 197 cdot 691 cdot 1123 cdot 6131 cdot mbox{BIG}
107 10284720757613717413913 = 3^3 cdot mbox{BIG}
108 16641027750620563662096 = 2^2 cdot 31 cdot 59 cdot 1394177 cdot mbox{BIG}
109 26925748508234281076009 = 3^3 cdot mbox{BIG}
110 43566776258854844738105 = 3^3 cdot 2239 cdot mbox{BIG}
111 70492524767089125814114 = 2^3 cdot 3 cdot 59753 cdot mbox{BIG}
112 114059301025943970552219 = 7 cdot 19 cdot mbox{BIG}
113 184551825793033096366333 = 3 cdot 781063 cdot 1419973 cdot mbox{BIG}
114 298611126818977066918552 = 2^2 cdot 3^2 cdot 139 cdot 337 cdot 1307 cdot mbox{BIG}
115 483162952612010163284885 = 3 cdot 137 cdot mbox{BIG}
116 781774079430987230203437 = 139^2 cdot mbox{BIG}
117 1264937032042997393488322 = 2^3 cdot 3 cdot mbox{BIG}
118 2046711111473984623691759 = 3^4 cdot 349187 cdot mbox{BIG}
I adjusted the Fermat command to also show the resulting number; in this case every other Lucas number
6 8 Fermat: ( 5 , 12 , 18 );
8 21 Fermat: ( 5 , 12 , 47 ); ( 20 , 24 , 94 );
10 55 Fermat: ( 5 , 12 , 123 ); ( 20 , 24 , 246 );
12 144 Fermat: ( 5 , 12 , 322 ); ( 20 , 24 , 644 );
14 377 Fermat: ( 5 , 12 , 843 ); ( 20 , 24 , 1686 );
16 987 Fermat: ( 5 , 12 , 2207 ); ( 20 , 24 , 4414 );
18 2584 Fermat: ( 5 , 12 , 5778 ); ( 20 , 24 , 11556 );
20 6765 Fermat: ( 5 , 12 , 15127 ); ( 20 , 24 , 30254 );
22 17711 Fermat: ( 5 , 12 , 39603 ); ( 20 , 24 , 79206 );
24 46368 Fermat: ( 5 , 12 , 103682 ); ( 20 , 24 , 207364 );
26 121393 Fermat: ( 5 , 12 , 271443 ); ( 20 , 24 , 542886 );
28 317811 Fermat: ( 5 , 12 , 710647 ); ( 20 , 24 , 1421294 );
30 832040 Fermat: ( 5 , 12 , 1860498 ); ( 20 , 24 , 3720996 );
32 2178309 Fermat: ( 5 , 12 , 4870847 ); ( 20 , 24 , 9741694 );
34 5702887 Fermat: ( 5 , 12 , 12752043 ); ( 20 , 24 , 25504086 );
36 14930352 Fermat: ( 5 , 12 , 33385282 ); ( 20 , 24 , 66770564 );
38 39088169 Fermat: ( 5 , 12 , 87403803 ); ( 20 , 24 , 174807606 );
40 102334155 Fermat: ( 5 , 12 , 228826127 ); ( 20 , 24 , 457652254 );
42 267914296 Fermat: ( 5 , 12 , 599074578 ); ( 20 , 24 , 1198149156 );
44 701408733 Fermat: ( 5 , 12 , 1568397607 ); ( 20 , 24 , 3136795214 );
46 1836311903 Fermat: ( 5 , 12 , 4106118243 ); ( 20 , 24 , 8212236486 );
48 4807526976 Fermat: ( 5 , 12 , 10749957122 ); ( 20 , 24 , 21499914244 );
50 12586269025 Fermat: ( 5 , 12 , 28143753123 ); ( 20 , 24 , 56287506246 );
52 32951280099 Fermat: ( 5 , 12 , 73681302247 ); ( 20 , 24 , 147362604494 );
54 86267571272 Fermat: ( 5 , 12 , 192900153618 ); ( 20 , 24 , 385800307236 );
56 225851433717 Fermat: ( 5 , 12 , 505019158607 ); ( 20 , 24 , 1010038317214 );
58 591286729879 Fermat: ( 5 , 12 , 1322157322203 ); ( 20 , 24 , 2644314644406 );
60 1548008755920 Fermat: ( 5 , 12 , 3461452808002 ); ( 20 , 24 , 6922905616004 );
62 4052739537881 Fermat: ( 5 , 12 , 9062201101803 ); ( 20 , 24 , 18124402203606 );
64 10610209857723 Fermat: ( 5 , 12 , 23725150497407 ); ( 20 , 24 , 47450300994814 );
66 27777890035288 Fermat: ( 5 , 12 , 62113250390418 ); ( 20 , 24 , 124226500780836 );
68 72723460248141 Fermat: ( 5 , 12 , 162614600673847 ); ( 20 , 24 , 325229201347694 );
70 190392490709135 Fermat: ( 5 , 12 , 425730551631123 ); ( 20 , 24 , 851461103262246 );
72 498454011879264 Fermat: ( 5 , 12 , 1114577054219522 ); ( 20 , 24 , 2229154108439044 );
74 1304969544928657 Fermat: ( 5 , 12 , 2918000611027443 ); ( 20 , 24 , 5836001222054886 );
76 3416454622906707 Fermat: ( 5 , 12 , 7639424778862807 ); ( 20 , 24 , 15278849557725614 );
78 8944394323791464 Fermat: ( 5 , 12 , 20000273725560978 ); ( 20 , 24 , 40000547451121956 );
80 23416728348467685 Fermat: ( 5 , 12 , 52361396397820127 ); ( 20 , 24 , 104722792795640254 );
Edit:
I'll keep my original at the bottom since it has some value under this question.
Without considering $n:F_n^2-28\lt 0$ (for example, $n=4,F_4^2-28=9-28=-19$ or similarly for $n=5$), the first couple values certainly are composite, but not in an immediately-obvious manner.
Here is an answer looking at the Fermat numbers, which are not under discussion for this question but are mildly interesting in this light anyway:
Disregarding $F_0=2^{2^0}+1$, we have $3\nmid 2^{2^n}+1$ for all positive integers $n$, making $F_n^2\equiv 1\pmod 3$, and as $28\equiv 1\pmod 3$ we have $F_n^2-28\equiv 0\pmod 3$ for all $n\gt 0$.