palindrome

The idea was to associate every letter of an alphabet to a number. The number would be converted in base K, where K is the number of letters in the alphabet. We then want to find palindromic words that, converted in base 10, have also a palindromic number in base 10.

Example:
The word KYRGRYK gives the number 5760110675 in base 10 and BQJ7JQB in base K, using the Spanish alphabet with the letters in alphabetic orders. Though, it does not have a meaning in Spanish. Maybe in other languages.

Finding meaningful words or phrases looks almost impossible. But who knows?

spanish letters	index	baseK
_	0	0
A	1	1
B	2	2
C	3	3
D	4	4
E	5	5
F	6	6
G	7	7
H	8	8
I	9	9
J	10	A
K	12	B
L	11	C
M	13	D
N	14	E
Ñ	15	F
O	16	G
P	17	H
Q	18	I
R	19	J
S	20	K
T	21	L
U	22	M
V	23	N
W	24	O
X	25	P
Y	26	Q
Z	27	R

Performance

Without being too much optimized and on a random computer such as mine, here are some speed results

max number	digits	time
10^6 = 1_000_000	7	~2 ms
10^9 = 1_000_000_000	10	~200 ms
10^12 = 1_000_000_000_000	13	~2 secs
10^15 = 1_000_000_000_000_000	16	~2 mins
10^18 = 1_000_000_000_000_000_000	19	~50 mins

Result

International Alphabet

Number in base K	Number in base 10	Word in International Alphabet (Letter in alphabetic order)	Word in International Alphabet (Order by most common letters)	Word in International Alphabet (Letter in alphabetic order with space at the end)
		_ABCDEFGHIJKLMNOPQRSTUVWXYZ	_EAISNRTOLUDCMPGBVHFQYXJKWZ	ABCDEFGHIJKLMNOPQRSTUVWXYZ_
0	0	_	_	A
1	1	A	E	B
2	2	B	A	C
3	3	C	I	D
4	4	D	S	E
5	5	E	N	F
6	6	F	R	G
7	7	G	T	H
8	8	H	O	I
9	9	I	L	J
B	11	K	D	L
M	22	V	X	W
99	252	II	LL	JJ
MM	616	VV	XX	WW
111	757	AAA	EEE	BBB
141	838	ADA	ESE	BEB
171	919	AGA	ETE	BHB
E3E	10301	NCN	PIP	ODO
HNH	13031	QWQ	VJV	RXR
LNL	15951	UWU	YJY	VXV
ODO	17871	XMX	KMK	YNY
3993	65856	CIIC	ILLI	DJJD
26N62	1197911	BFWFB	ARJRA	CGXGC
48684	2287822	DHFHD	SOROS	EIGIE
86M68	4385834	HFVFH	ORXRO	IGWGI
A858A	5475745	JHEHJ	UONOU	KIFIK
ABPBA	5549455	JKYKJ	UDWDU	KLZLK
BLQLB	6278726	KUZUK	DYZYD	LV_VL
CD8DC	6639366	LMHML	CMOMC	MNINM
DN9ND	7368637	MWIWM	MJLJM	NXJXN
E6L6E	7573757	NFUFN	PRYRP	OGVGO
G848G	8663668	PHDHP	BOSOB	QIEIQ
GBOBG	8737378	PKXKP	BDKDB	QLYLQ
HI5IH	9392939	QRERQ	VHNHV	RSFSR
HLPLH	9466649	QUYUQ	VYWYV	RVZVR
ID7DI	9827289	RMGMR	HMTMH	SNHNS
4IAAI4	67166176	DRJJRD	SHUUHS	ESKKSE
EP00PE	214171412	NY__YN	PW__WP	OZAAZO
1FDCDF1	609808906	AOMLMOA	EGMCMGE	BPNMNPB
2474742	836040638	BDGDGDB	ASTSTSA	CEHEHEC
5DHAHD5	2132882312	EMQJQME	NMVUVMN	FNRKRNF
6BAGAB6	2487997842	FKJPJKF	RDUBUDR	GLKQKLG
DMMJMMD	5364224635	MVVSVVM	MXXFXXM	NWWTWWN
LJGOGJL	8417447148	USPXPSU	YFBKBFY	VTQYQTV
NA7N7AN	9058338509	WJGWGJW	JUTJTUJ	XKHXHKX
9QPP4PPQ9	2823881883282	IZYYDYYZI	LZWWSWWZL	J_ZZEZZ_J
A7IJ8JI7A	2904768674092	JGRSHSRGJ	UTHFOFHTU	KHSTITSHK
ADMG6GMDA	2969036309692	JMVPFPVMJ	UMXBRBXMU	KNWQGQWNK
DDPL7LPDD	3817559557183	MMYUGUYMM	MMWYTYWMM	NNZVHVZNN
E0B474B0E	3958336338593	N_KDGDK_N	P_DSTSD_P	OALEHELAO
F450O054F	4280234320824	ODE_X_EDO	GSN_K_NSG	PEFAYAFEP
FK1P6P1KF	4446399936444	OTAYFYATO	GQEWRWEQG	PUBZGZBUP
G04D0D40G	4520609060254	P_DM_MD_P	B_SM_MS_B	QAENANEAQ
GJLF4FLJG	4725972795274	PSUODOUSP	BFYGSGYFB	QTVPEPVTQ
H0E989E0H	4806859586084	Q_NIHIN_Q	V_PLOLP_V	RAOJIJOAR
HNQJNJQNH	5052248422505	QWZSWSZWQ	VJZFJFZJV	RX_TXT_XR
IAC2N2CAI	5193025203915	RJLBWBLJR	HUCAJACUH	SKMCXCMKS
KCE030ECK	5779540459775	TLN_C_NLT	QCP_I_PCQ	UMOADAOMU
L7PPEPP7L	6014294924106	UGYYNYYGU	YTWWPWWTY	VHZZOZZHV
LAF7O7FAL	6041548451406	UJOGXGOJU	YUGTKTGUY	VKPHYHPKV
2KNIH1HINK2	570984989489075	BTWRQAQRWTB	AQJHVEVHJQA	CUXSRBRSXUC
403Q464Q304	824685424586428	D_CZDFDZC_D	S_IZSRSZI_S	EAD_EGE_DAE
46CN9I9NC64	872951606159278	DFLWIRIWLFD	SRCJLHLJCRS	EGMXJSJXMGE
4I8CJ5JC8I4	963217686712369	DRHLSESLHRD	SHOCFNFCOHS	ESIMTFTMISE
4LC28982CL4	987115383511789	DULBHIHBLUD	SYCAOLOACYS	EVMCIJICMVE
659CHHHC956	1276149009416721	FEILQQQLIEF	RNLCVVVCLNR	GFJMRRRMJFG
C87JG8GJ78C	2533880440883352	LHGSPHPSGHL	COTFBOBFTOC	MIHTQIQTHIM
I8PG050GP8I	3774273333724773	RHYP_E_PYHR	HOWB_N_BWOH	SIZQAFAQZIS
BA44ACCA44AB	63240319091304236	KJDDJLLJDDJK	DUSSUCCUSSUD	LKEEKMMKEEKL
1Q60LFAFL06Q1	295871648846178592	AZF_UOJOU_FZA	EZR_YGUGY_RZE	B_GAVPKPVAG_B
3GDCEIDIECDG3	542001114445237365	CPMLNRMRNLMPC	IBMCPHMHPCMBI	DQNMOSNSOMNQD
507CB4B4BC705	752009074470900257	E_GLKDKDKLG_E	N_TCDSDSDCT_N	FAHMLELELMHAF
6A42DMLMD24A6	957001143341100759	FJDBMVUVMBDJF	RUSAMXYXMASUR	GKECNWVWNCEKG

Spanish Alphabet

Number in base K	Number in base 10	Word in Spanish Alphabet (Letter in alphabetic order)	Word in Spanish Alphabet (Vowel in even position)	Word in Spanish Alphabet (Vowel in odd position)	Word in Spanish Alphabet (Order by most common letters)	Word in Spanish Alphabet (Vowels first)	Word in Spanish Alphabet (Order random)	Word in Spanish Alphabet (Letter in alphabetic order with space at the end)
		_ABCDEFGHIJKLMNÑOPQRSTUVWXYZ	_BACDFEGHJIKLMNÑOPQRSTUVWXYZ	_ABCDEFGHIJKLMNOÑPQRSUTVWYXZ	_EAOSRNIDLCTUMPBGVYQHFZJÑXKW	_EAOIUNSDLCTRMPBGVYQHFZJÑXKW	_EAFIUNSDLCTRMPBGJYQHOZVÑXKW	ABCDEFGHIJKLMNÑOPQRSTUVWXYZ_
0	0	_	_	_	_	_	_	A
1	1	A	B	A	E	E	E	B
2	2	B	A	B	A	A	A	C
3	3	C	C	C	O	O	F	D
4	4	D	D	D	S	I	I	E
5	5	E	F	E	R	U	U	F
6	6	F	E	F	N	N	N	G
7	7	G	G	G	I	S	S	H
8	8	H	H	H	D	D	D	I
9	9	I	J	I	L	L	L	J
B	11	K	K	K	T	T	T	L
M	22	U	U	T	Z	Z	Z	V
88	232	HH	HH	HH	DD	DD	DD	II
GG	464	OO	OO	ÑÑ	GG	GG	GG	PP
OO	696	WW	WW	WW	ÑÑ	ÑÑ	ÑÑ	XX
7A7	5775	GJG	GIG	GJG	ICI	SCS	SCS	HKH
828	6336	HBH	HAH	HBH	DAD	DAD	DAD	ICI
D7D	10401	MGM	MGM	MGM	MIM	MSM	MSM	NHN
JCJ	15251	RLR	RLR	RLR	QUQ	QRQ	QRQ	SMS
LHL	16961	TPT	TPT	UPU	FVF	FVF	OJO	UQU
QEQ	20802	YNY	YNY	XNX	KPK	KPK	KPK	ZÑZ
4NBN4	2972792	DVKVD	DVKVD	DVKVD	SJTJS	IJTJI	IVTVI	EWLWE
97D79	5695965	IGMGI	JGMGJ	IGMGI	LIMIL	LSMSL	LSMSL	JHNHJ
A3O3A	6231326	JCWCJ	ICWCI	JCWCJ	COÑOC	COÑOC	CFÑFC	KDXDK
AKJKA	6601066	JSRSJ	ISRSI	JSRSJ	CHQHC	CHQHC	CHQHC	KTSTK
BB0BB	7003007	KK_KK	KK_KK	KK_KK	TT_TT	TT_TT	TT_TT	LLALL
E8A8E	8788878	NHJHN	NHIHN	NHJHN	PDCDP	PDCDP	PDCDP	ÑIKIÑ
F4L4F	9324239	ÑDTDÑ	ÑDTDÑ	ODUDO	BSFSB	BIFIB	BIOIB	OEUEO
H282H	10499401	PBHBP	PAHAP	PBHBP	VADAV	VADAV	JADAJ	QCICQ
9A00A9	161040161	IJ__JI	JI__IJ	IJ__JI	LC__CL	LC__CL	LC__CL	JKAAKJ
ONQQNO	427777724	WVYYVW	WVYYVW	WVXXVW	ÑJKKJÑ	ÑJKKJÑ	ÑVKKVÑ	XWZZWX
112C211	500595005	AABLBAA	BBALABB	AABLBAA	EEAUAEE	EEARAEE	EEARAEE	BBCMCBB
1D9R9D1	711757117	AMIZIMA	BMJZJMB	AMIZIMA	EMLWLME	EMLWLME	EMLWLME	BNJ_JNB
1FILIF1	751585157	AÑQTQÑA	BÑQTQÑB	AOQUQOA	EBYFYBE	EBYFYBE	EBYOYBE	BORUROB
1GR3RG1	773939377	AOZCZOA	BOZCZOB	AÑZCZÑA	EGWOWGE	EGWOWGE	EGWFWGE	BP_D_PB
1OF0FO1	904171409	AWÑ_ÑWA	BWÑ_ÑWB	AWO_OWA	EÑB_BÑE	EÑB_BÑE	EÑB_BÑE	BXOAOXB
1R0Q0R1	947141749	AZ_Y_ZA	BZ_Y_ZB	AZ_X_ZA	EW_K_WE	EW_K_WE	EW_K_WE	B_AZA_B
35B8B53	1538668351	CEKHKEC	CFKHKFC	CEKHKEC	ORTDTRO	OUTDTUO	FUTDTUF	DFLILFD
BQJ7JQB	5760110675	KYRGRYK	KYRGRYK	KXRGRXK	TKQIQKT	TKQSQKT	TKQSQKT	LZSHSZL
CMM7MMC	6175005716	LUUGUUL	LUUGUUL	LTTGTTL	UZZIZZU	RZZSZZR	RZZSZZR	MVVHVVM
F41E14F	7298118927	ÑDANADÑ	ÑDBNBDÑ	ODANADO	BSEPESB	BIEPEIB	BIEPEIB	OEBÑBEO
LOR6ROL	10549494501	TWZFZWT	TWZEZWT	UWZFZWU	FÑWNWÑF	FÑWNWÑF	OÑWNWÑO	UX_G_XU
N96L69N	11242524211	VIFTFIV	VJETEJV	VIFUFIV	JLNFNLJ	JLNFNLJ	VLNONLV	WJGUGJW
RD8E8DR	13240004231	ZMHNHMZ	ZMHNHMZ	ZMHNHMZ	WMDPDMW	WMDPDMW	WMDPDMW	_NIÑIN_
RKCNCKR	13363136331	ZSLVLSZ	ZSLVLSZ	ZSLVLSZ	WHUJUHW	WHRJRHW	WHRVRHW	_TMWMT_
5O5DD5O5	79124342197	EWEMMEWE	FWFMMFWF	EWEMMEWE	RÑRMMRÑR	UÑUMMUÑU	UÑUMMUÑU	FXFNNFXF
6H7EE7H6	89279097298	FPGNNGPF	EPGNNGPE	FPGNNGPF	NVIPPIVN	NVSPPSVN	NJSPPSJN	GQHÑÑHQG
KD5HH5DK	276220022672	SMEPPEMS	SMFPPFMS	SMEPPEMS	HMRVVRMH	HMUVVUMH	HMUJJUMH	TNFQQFNT
11A414A11	396183381693	AAJDADJAA	BBIDBDIBB	AAJDADJAA	EECSESCEE	EECIEICEE	EECIEICEE	BBKEBEKBB
1Q5AKA5Q1	731212212137	AYEJSJEYA	BYFISIFYB	AXEJSJEXA	EKRCHCRKE	EKUCHCUKE	EKUCHCUKE	BZFKTKFZB
6RA555AR6	2636029206362	FZJEEEJZF	EZIFFFIZE	FZJEEEJZF	NWCRRRCWN	NWCUUUCWN	NWCUUUCWN	G_KFFFK_G
9EI4G4IE9	3597871787953	INQDODQNI	JNQDODQNJ	INQDÑDQNI	LPYSGSYPL	LPYIGIYPL	LPYIGIYPL	JÑREPERÑJ
APBDBDBPA	4120874780214	JXKMKMKXJ	IXKMKMKXI	JYKMKMKYJ	CXTMTMTXC	CXTMTMTXC	CXTMTMTXC	KYLNLNLYK
CC8J1J8CC	4699722279964	LLHRARHLL	LLHRBRHLL	LLHRARHLL	UUDQEQDUU	RRDQEQDRR	RRDQEQDRR	MMISBSIMM
F910C019F	5788955598875	ÑIA_L_AIÑ	ÑJB_L_BJÑ	OIA_L_AIO	BLE_U_ELB	BLE_R_ELB	BLE_R_ELB	OJBAMABJO
GP62L26PG	6385093905836	OXFBTBFXO	OXEATAEXO	ÑYFBUBFYÑ	GXNAFANXG	GXNAFANXG	GXNAOANXG	PYGCUCGYP
M1EHRHE1M	8332192912338	UANPZPNAU	UBNPZPNBU	TANPZPNAT	ZEPVWVPEZ	ZEPVWVPEZ	ZEPJWJPEZ	VBÑQ_QÑBV
M1RAIAR1M	8338331338338	UAZJQJZAU	UBZIQIZBU	TAZJQJZAT	ZEWCYCWEZ	ZEWCYCWEZ	ZEWCYCWEZ	VB_KRK_BV
PICF3FCIP	9693965693969	XQLÑCÑLQX	XQLÑCÑLQX	YQLOCOLQY	XYUBOBUYX	XYRBOBRYX	XYRBFBRYX	YRMODOMRY
53APRRPA35	54173144137145	ECJXZZXJCE	FCIXZZXICF	ECJYZZYJCE	ROCXWWXCOR	UOCXWWXCOU	UFCXWWXCFU	FDKY__YKDF
124HGGGH421	319102262201913	ABDPOOOPDBA	BADPOOOPDAB	ABDPÑÑÑPDBA	EASVGGGVSAE	EAIVGGGVIAE	EAIJGGGJIAE	BCEQPPPQECB
1LIJFQFJIL1	525408828804525	ATQRÑYÑRQTA	BTQRÑYÑRQTB	AUQROXORQUA	EFYQBKBQYFE	EFYQBKBQYFE	EOYQBKBQYOE	BURSOZOSRUB
1NOG363GON1	548785939587845	AVWOCFCOWVA	BVWOCECOWVB	AVWÑCFCÑWVA	EJÑGONOGÑJE	EJÑGONOGÑJE	EVÑGFNFGÑVE	BWXPDGDPXWB
23ED5H5DE32	629596242695926	BCNMEPEMNCB	ACNMFPFMNCA	BCNMEPEMNCB	AOPMRVRMPOA	AOPMUVUMPOA	AFPMUJUMPFA	CDÑNFQFNÑDC
2OA5EDE5AO2	850128939821058	BWJENMNEJWB	AWIFNMNFIWA	BWJENMNEJWB	AÑCRPMPRCÑA	AÑCUPMPUCÑA	AÑCUPMPUCÑA	CXKFÑNÑFKXC
32MOQRQOM23	918395696593819	CBUWYZYWUBC	CAUWYZYWUAC	CBTWXZXWTBC	OAZÑKWKÑZAO	OAZÑKWKÑZAO	FAZÑKWKÑZAF	DCVXZ_ZXVCD
HK82EEE28KH	5249970550799425	PSHBNNNBHSP	PSHANNNAHSP	PSHBNNNBHSP	VHDAPPPADHV	VHDAPPPADHV	JHDAPPPADHJ	QTICÑÑÑCITQ
2C6E7117E6C2	20210235053201202	BLFNGAAGNFLB	ALENGBBGNELA	BLFNGAAGNFLB	AUNPIEEIPNUA	ARNPSEESPNRA	ARNPSEESPNRA	CMGÑHBBHÑGMC
19PG2HNH2GP91	314435021120534413	AIXOBPVPBOXIA	BJXOAPVPAOXJB	AIYÑBPVPBÑYIA	ELXGAVJVAGXLE	ELXGAVJVAGXLE	ELXGAJVJAGXLE	BJYPCQWQCPYJB

Follow-ups

To improve the search, it needs to improve the way we find palindromic numbers. It's already good, but not perfect.

Also, we can try to optimize even better the computation accross threads.