## Sunday, 24 March 2013

### SUNDAY, 24 MARCH 2013

Today is the $83^{rd}$ day of the year.

$83$ is a prime number. In fact $(2 \times 83) + 1 = 167$ is prime which means that $83$ is a Sophie Germain prime, see A005384.
Given that
$20$ is not prime and $(2 \times 20) + 1 = 41$
$41$ is prime and $(2 \times 41) + 1 = 83$
$83$ is prime and $(2 \times 83) + 1 = 167$
$167$ is prime and $(2 \times 167) + 1 = 335$
$335$ is not prime
Then we have a Cunningham Chain of length $3$ i.e. $(41, 83, 167)$, see A059762.

Take every number less than or equal to $83$, write it down in base $25$, something like this:

$1_{10} = 1_{25}$
$2_{10} = 2_{25}$
$3_{10} = 3_{25}$
$4_{10} = 4_{25}$
$5_{10} = 5_{25}$
$6_{10} = 6_{25}$
$7_{10} = 7_{25}$
$8_{10} = 8_{25}$
$9_{10} = 9_{25}$
$10_{10} = a_{25}$
$11_{10} = b_{25}$
$12_{10} = c_{25}$
$13_{10} = d_{25}$
$14_{10} = e_{25}$
$15_{10} = f_{25}$
$16_{10} = g_{25}$
$17_{10} = h_{25}$
$18_{10} = i_{25}$
$19_{10} = j_{25}$
$20_{10} = k_{25}$
$21_{10} = l_{25}$
$22_{10} = m_{25}$
$23_{10} = n_{25}$
$24_{10} = o_{25}$
$25_{10} = 10_{25}$
$26_{10} = 11_{25}$
$27_{10} = 12_{25}$
$28_{10} = 13_{25}$
...
$59_{10} = 29_{25}$
$60_{10} = 2a_{25}$
$61_{10} = 2b_{25}$
$62_{10} = 2c_{25}$
$63_{10} = 2d_{25}$
$64_{10} = 2e_{25}$
$65_{10} = 2f_{25}$
$66_{10} = 2g_{25}$
$67_{10} = 2h_{25}$
$68_{10} = 2i_{25}$
$69_{10} = 2j_{25}$
$70_{10} = 2k_{25}$
$71_{10} = 2l_{25}$
$72_{10} = 2m_{25}$
$73_{10} = 2n_{25}$
$74_{10} = 2o_{25}$
$75_{10} = 30_{25}$
$76_{10} = 31_{25}$
$77_{10} = 32_{25}$
$78_{10} = 33_{25}$
$79_{10} = 34_{25}$
$80_{10} = 35_{25}$
$81_{10} = 36_{25}$
$82_{10} = 37_{25}$
$83_{10} = 38_{25}$

Now, concatenate each of the base $25$ values starting with $1$ and reversing each value before it is concatenated giving:
$1,234,567,89a,bcd,efg,hij,klm,no0,111,213,141,516,171,819,1a1,b1c,1d1,e1f,1g1,h1i,1j1,k1l,1m1,n1o,102,122,232,425,262,728,292,a2b,2c2,d2e,2f2,g2h,2i2,j2k,2l2,m2n,2o2,031,323,334,353,637,383$
Convert this $142$ digit number in base $25$ to decimal and the result is divisible by $8$.
Not in itself remarkable but given that the number $83$ is the lowest number that can generate a multiple of $8$ in this fashion is remarkable, see A029518.

Consider, the sum of the squares of all the numbers up to and including $83$. This is $194,054$ which is divisible by $83$, see A007310.

 Square Total $1^2 = 1$ $1$ $2^2 = 4$ $5$ $3^2 = 9$ $14$ $4^2 = 16$ $30$ $5^2 = 25$ $55$ $6^2 = 36$ $91$ $7^2 = 49$ $140$ $8^2 = 64$ $204$ $9^2 = 81$ $285$ $10^2 = 100$ $385$ $11^2 = 121$ $506$ $12^2 = 144$ $650$ $13^2 = 169$ $819$ $14^2 = 196$ $1015$ $15^2 = 225$ $1240$ $16^2 = 256$ $1496$ $17^2 = 289$ $1785$ $18^2 = 324$ $2109$ $19^2 = 361$ $2470$ $20^2 = 400$ $2870$ $21^2 = 441$ $3311$ $22^2 = 484$ $3795$ $23^2 = 529$ $4324$ $24^2 = 576$ $4900$ $25^2 = 625$ $5525$ $26^2 = 676$ $6201$ $27^2 = 729$ $6930$ $28^2 = 784$ $7714$ $29^2 = 841$ $8555$ $30^2 = 900$ $9455$ $31^2 = 961$ $10416$ $32^2 = 1024$ $11440$ $33^2 = 1089$ $12529$ $34^2 = 1156$ $13685$ $35^2 = 1225$ $14910$ $36^2 = 1296$ $16206$ $37^2 = 1369$ $17575$ $38^2 = 1444$ $19019$ $39^2 = 1521$ $20540$ $40^2 = 1600$ $22140$ $41^2 = 1681$ $23821$ $42^2 = 1764$ $25585$ $43^2 = 1849$ $27434$ $44^2 = 1936$ $29370$ $45^2 = 2025$ $31395$ $46^2 = 2116$ $33511$ $47^2 = 2209$ $35720$ $48^2 = 2304$ $38024$ $49^2 = 2401$ $40425$ $50^2 = 2500$ $42925$ $51^2 = 2601$ $45526$ $52^2 = 2704$ $48230$ $53^2 = 2809$ $51039$ $54^2 = 2916$ $53955$ $55^2 = 3025$ $56980$ $56^2 = 3136$ $60116$ $57^2 = 3249$ $63365$ $58^2 = 3364$ $66729$ $59^2 = 3481$ $70210$ $60^2 = 3600$ $73810$ $61^2 = 3721$ $77531$ $62^2 = 3844$ $81375$ $63^2 = 3969$ $85344$ $64^2 = 4096$ $89440$ $65^2 = 4225$ $93665$ $66^2 = 4356$ $98021$ $67^2 = 4489$ $102510$ $68^2 = 4624$ $107134$ $69^2 = 4761$ $111895$ $70^2 = 4900$ $116795$ $71^2 = 5041$ $121836$ $72^2 = 5184$ $127020$ $73^2 = 5329$ $132349$ $74^2 = 5476$ $137825$ $75^2 = 5625$ $143450$ $76^2 = 5776$ $149226$ $77^2 = 5929$ $155155$ $78^2 = 6084$ $161239$ $79^2 = 6241$ $167480$ $80^2 = 6400$ $173880$ $81^2 = 6561$ $180441$ $82^2 = 6724$ $187165$ $83^2 = 6889$ $194054$